Hindawi Publishing CorporationJournal of Discrete MathematicsVolume 2013 Article ID 170263 13 pageshttpdxdoiorg1011552013170263

Research ArticleFinite Iterative Algorithm for Solving a Complex ofConjugate and Transpose Matrix Equation

Mohamed A Ramadan1 Talaat S El-Danaf1 and Ahmed M E Bayoumi2

1 Department of Mathematics Faculty of Science Menoufia University Shebeen El-Koom Egypt2 Department of Mathematics Faculty of Education Ain Shams University Cairo Egypt

Correspondence should be addressed to Mohamed A Ramadan mramadaneuneg

Received 4 August 2012 Accepted 4 November 2012

Academic Editor Franck Petit

Copyright copy 2013 Mohamed A Ramadan et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We consider an iterative algorithm for solving a complex matrix equation with conjugate and transpose of two unknowns of theform 119860

11198811198611+ 11986211198821198631+ 11986021198811198612+ 11986221198821198632+ 11986031198811198671198613+ 11986231198821198671198633+ 11986041198811198791198614+ 11986241198821198791198634= 119864 With the iterative algorithm the

existence of a solution of this matrix equation can be determined automatically When this matrix equation is consistent for anyinitial matrices 119881

1 1198821the solutions can be obtained by iterative algorithm within finite iterative steps in the absence of round-off

errors Some lemmas and theorems are stated and proved where the iterative solutions are obtained A numerical example is givento illustrate the effectiveness of the proposed method and to support the theoretical results of this paper

1 Introduction

Consider the complex matrix equation

11986011198811198611+ 11986211198821198631+ 11986021198811198612+ 11986221198821198632

+ 11986031198811198671198613+ 11986231198821198671198633+ 11986041198811198791198614+ 11986241198821198791198634= 119864

(1)

where 1198601 1198602 1198621 1198622

isin C119898times119903 1198611 1198612 1198631 1198632

isin C119904times1198991198603 1198604 1198623 1198624isin C119898times119904 119864 isin C119898times119899 and 119861

3 1198614 1198633 1198634isin

C119903times119899 are givenmatrices while119881 119882 isin C119903times119904 arematrices to bedetermined In the field of linear algebra iterative algorithmsfor solving matrix equations have received much attentionBased on the iterative solutions ofmatrix equations Ding andChen presented the hierarchical gradient iterative algorithmsfor general matrix equations [1 2] and hierarchical leastsquares iterative algorithms for generalized coupled Sylvestermatrix equations and general coupled matrix equations [34] The hierarchical gradient iterative algorithms [1 2] andhierarchical least squares iterative algorithms [1 4 5] for solv-ing general (coupled) matrix equations are innovational andcomputationally efficient numerical ones and were proposedbased on the hierarchical identification principle [3 6] which

regards the unknown matrix as the system parameter matrixto be identified Iterative algorithms were proposed for con-tinuous and discrete Lyapunov matrix equations by applyingthe hierarchical identification principle [7] Recently the ideaof the hierarchical identification was also utilized to solvethe so-called extended Sylvester-conjugate matrix equationsin [8] From an optimization point of view a gradient-basediteration was constructed in [9] to solve the general coupledmatrix equation A significant feature of the method in [9]is that a necessary and sufficient condition guaranteeing theconvergence of the algorithm can be explicitly obtained

Some complex matrix equations have attracted attentionfrom many researchers since it was shown in [10] that theconsistence of the matrix equation 119860119883 minus 119883119861 = 119862 can becharacterized by the consimilarity [11ndash13] of two partitionedmatrices related to the coefficient matrices 119860 119861 and 119862 Byconsimilarity Jordan decomposition explicit solutions wereobtained in [10 14] Some explicit expressions of the solutionto the matrix equation 119860119883 minus 119883119861 = 119862 were established in[15] and it was shown that this matrix equation has a uniquesolution if and only if 119860119860 and 119861119861 hav no common eigen-values Research on solving linear matrix equations has beenactively engaged in formany years For exampleNavarra et al

2 Journal of Discrete Mathematics

studied a representation of the general common solution ofthe matrix equations 119860

11198831198611

= 1198621and 119860

21198831198612

= 1198622[16]

Van derWoude obtained the existence of a common solution119883 for thematrix equations119860

119894119883119861119895= 119862119894119895[17] Bhimasankaram

considered the linear matrix equations 119860119883 = 119861 119862119883 = 119863and 119864119883119865 = 119866 [18] Mitra has provided conditions for theexistence of a solution and a representation of the generalcommon solution of thematrix equations119860119883 = 119862 and 119883119861 =

119863 and the matrix equations 11986011198831198611= 1198621and 119860

21198831198612= 1198622

[19 20] Ramadan et al [21] introduced a complete generaland explicit solution to the Yakubovich matrix equation 119881 minus

119860119881119865 = 119861119882 and the matrix equation (119860119883119861 119866119883119867) = (119862119863)

has some important results that have been developed In [22]necessary and sufficient conditions for its solvability and theexpression of the solution were derived by means of gener-alized inverse Moreover in [22] the least-squares solutionwas also obtained by using the generalized singular valuedecomposition While in [23] when this matrix equation isconsistent the minimum-norm solution was given by the useof the canonical correlation decomposition In [24] basedon the projection theorem in Hilbert space an analyticalexpression of the least-squares solution was given for thematrix equation (119860119883119861 119866119883119867) = (119862119863) by making use of thegeneralized singular value decomposition and the canonicalcorrelation decomposition In [25] by using the matrix rankmethod a necessary and sufficient condition was derivedfor the matrix equations 119860119883

1119861 = 119862 and 119866119883

2119867 = 119863 to

have a common least square solution In the aforementionedmethods the coefficient matrices of the considered equationsare required to be firstly transformed into some canonicalforms Recently an iterative algorithm has presented in [26]to solve thematrix equation (119860119883119861 119866119883119867) = (119862119863) Differentfrom the above mentioned methods this algorithm can beimplemented by initial coefficient matrices and can provide asolution within finite iteration steps for any initial values

Very recently in [27] a new operator of conjugate productfor complex polynomialmatrices is proposed It is shown thatan arbitrary complex polynomial matrix can be convertedinto the so-called Smith normal form by elementary transfor-mations in the framework of conjugate product Meanwhilethe conjugate product and the Sylvester-conjugate sum arealso proposed in [28] Based on the important properties ofthe above new operators a unified approach to solve a generalclass of Sylvester-polynomial-conjugate matrix equations isgiven The complete solution of the Sylvester-polynomial-conjugate matrix equation is obtained In [29] by using a realinner product in complex matrix spaces a solution can beobtained within finite iterative steps for any initial values inthe absence of round-off errors In [30] iterative solutions toa class of complex matrix equations are given by applying thehierarchical identification principle

This paper is organized as follows First in Section 2we introduce some notations a lemma and a theoremthat will be needed to develop this work In Section 3 wepropose iterativemethods to obtain numerical solution to thecomplexmatrix equationwith conjugate and transpose of twounknowns of the form119860

11198811198611+11986211198821198631+11986021198811198612+11986221198821198632+

11986031198811198671198613+11986231198821198671198633+11986041198811198791198614+11986241198821198791198634= 119864 using iterative

method In Section 4 numerical example is given to explorethe simplicity and the neatness of the presented methods

2 Preliminaries

The following notations definitions lemmas and theo-rems will be used to develop the proposed work We use119860119879 119860 119860

119867 tr(119860) and 119860 to denote the transpose conjugateconjugate transpose the trace and the Frobenius norm of amatrix119860 respectively We denote the set of all119898times119899 complexmatrices byC119898times119899 and Re(119886) denote the real part of number 119886

Definition 1 (inner product [31]) A real inner product spaceis a vector space119881 over the real fieldR together with an innerproduct That is with a map

⟨sdot sdot⟩ 119881 times 119881 997888rarr R (2)

Satisfying the following three axioms for all vectors 119909 119910 119911 isin

119881 and all scalars 119886 isin R

(1) symmetry ⟨119909 119910⟩ = ⟨119910 119909⟩

(2) linearity in the first argument

⟨119886119909 119910⟩ = 119886 ⟨119909 119910⟩

⟨119909 + 119910 119911⟩ = ⟨119909 119911⟩ + ⟨119910 119911⟩

(3)

(3) positive definiteness ⟨119909 119909⟩ gt 0 for all 119909 = 0

two vectors 119906 119907 isin 119881 are said to be orthogonal if ⟨119906 119907⟩ =

0The following theorem defines a real inner product on

space C119898times119899 over the field R

Theorem 2 (see [32]) In the space C119898times119899 over the field R aninner product can be defined as

⟨119860 119861⟩ = Re [tr (119860119867119861)] (4)

3 The Main Result

In this section we propose an iterative solution to a com-plex matrix equation with conjugate and transpose of twounknowns

11986011198811198611+ 11986211198821198631+ 11986021198811198612+ 11986221198821198632+ 11986031198811198671198613

+ 11986231198821198671198633+ 11986041198811198791198614+ 11986241198821198791198634= 119864

(5)

defined in (1) where 1198601 1198602 1198621 1198622

isin C119898times119903 1198611 1198612 1198631

1198632isin C119904times119899 119860

3 1198604 1198623 1198624isin C119898times119904 119864 isin C119898times119899 and 119861

3 1198614

1198633 1198634isin C119903times119899 are given matrices while 119881 119882 isin C119903times119904 are

matrices to be determinedThe following finite iterative algorithm is presented to

solve it

Algorithm 3 (1) Input1198601119860211986211198622 1198611 11986121198631119863211986031198604

1198623 1198624 1198613 119861411986331198634 119864

(2) Chosen arbitrary matrices 11988111198821isin C119903times119904

Journal of Discrete Mathematics 3

(3) Set

1198771= 119864 minus 119860

111988111198611minus 119862111988211198631minus 119860211988111198612

minus 119862211988211198632minus 1198603119881119867

11198613minus 1198623119882119867

11198633

minus 1198604119881119879

11198614minus 1198624119882119879

11198634

1198751= 119860119867

11198771119861119867

1+ 119860119867

21198771119861119867

2+ 1198613119877119867

11198603+ 1198614119877119879

11198604

1198761= 119862119867

11198771119863119867

1+ 119862119867

21198771119863119867

2+ 1198633119877119867

11198623+ 1198634119877119879

11198624

119896 = 1

(6)

(4) If 119877119896= 0 then stop else go to Step 5

(5) Set

119881119896+1

= 119881119896+

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2119875119896

119882119896+1

= 119882119896+

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2119876119896

119877119896+1

= 119864 minus 1198601119881119896+1

1198611minus 1198621119882119896+1

1198631minus 1198602119881119896+1

1198612

minus 1198622119882119896+1

1198632minus 1198603119881119867

119896+11198613minus 1198623119882119867

119896+11198633

minus 1198604119881119879

119896+11198614minus 1198624119882119879

119896+11198634

119875119896+1

= 119860119867

1119877119896+1

119861119867

1+ 119860119867

2119877119896+1

119861119867

2+ 1198613119877119867

119896+11198603

+ 1198614119877119879

119896+11198604+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2119875119896

119876119896+1

= 119862119867

1119877119896+1

119863119867

1+ 119862119867

2119877119896+1

119863119867

2

+ 1198633119877119867

119896+11198623+ 1198634119877119879

119896+11198624+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2119876119896

(7)

(6) If 119877119896+1

= 0 then stop else let 119896 = 119896 + 1 go to Step 5To prove the convergence property of Algorithm 3 we

first establish the following basic properties

Lemma 4 Suppose that the matrix equation (1) is consistentand 119881

lowast 119882lowast are arbitrary solutions of (1) Then for any initialmatrices 119881

1and119882

1 we have

Re tr [119875119867119894

(119881lowastminus 119881119894) + 119876119867

119894(119882lowastminus 119882119894)] =

10038171003817100381710038171198771198941003817100381710038171003817

2

(8)

where the sequence 119881119894 119875119894 119882119894 119876119894 and 119877

119894 are gener-

ated by Algorithm 3 for 119894 = 1 2

Proof We applymathematical induction to prove the conclu-sion

For 119894 = 1 from Algorithm 3 we have

Re tr [1198751198671

(119881lowastminus 1198811) + 119876119867

1(119882lowastminus 1198821)]

= Re tr [(11986011986711198771119861119867

1+ 119860119867

21198771119861119867

2

+ 1198613119877119867

11198603+ 1198614119877119879

11198604119861119867

2)

119867

(119881lowastminus 1198811)

+ (119862119867

11198771119863119867

1+ 119862119867

21198771119863119867

2

+1198633119877119867

11198623+ 1198634119877119879

11198624119863119867

2)

119867

times (119882lowastminus 1198821) ]

= Re tr [11987711986711198601(119881lowastminus 1198811) 1198611+ 1198771

119867

1198602(119881lowastminus 1198811) 1198612

+ 1198771119861119867

3(119881lowastminus 1198811) 119860119867

3+ 1198771119861119867

4(119881lowastminus 1198811) 119860119867

4

+ 119877119867

11198621(119882lowastminus 1198821)1198631+ 1198771

119867

1198622(119882lowastminus 1198821)1198632

+ 1198771119863119867

3(119882lowastminus 1198821) 119862119867

3

+ 1198771119863119867

4(119882lowastminus 1198821) 119862119867

4]

(9)

From properties of trace and conjugate

Re tr [1198751198671

(119881lowastminus 1198811) + 119876119867

1(119882lowastminus 1198821)]

= Retr [11987711986711198601(119881lowastminus 1198811) 1198611+ 1198771

119867

1198602(119881lowast minus 119881

1) 1198612

+ 1198603(119881lowastminus 1198811)119867

1198613119877119867

1+ 1198604(119881lowastminus 1198811)119879

1198614119877119867

1

+ 119877119867

11198621(119882lowastminus 1198821)1198631

+ 1198771

119867

1198622(119882lowast minus 119882

1)1198632

+ 1198623(119882lowastminus 1198821)119867

1198633119877119867

1

+ 1198624(119882lowastminus 1198821)119879

1198634119877119867

1]

= Re tr [11987711986711198601(119881lowastminus 1198811) 1198611+ 119877119867

11198602(119881lowast minus 119881

1)1198612

+ 119877119867

11198603(119881lowastminus 1198811)119867

1198613

+ 119877119867

11198604(119881lowastminus 1198811)119879

1198614

+ 119877119867

11198621(119882lowastminus 1198821)1198631

+ 119877119867

11198622(119882lowast minus 119882

1)1198632

+ 119877119867

11198623(119882lowastminus 1198821)119867

1198633

+ 119877119867

11198624(119882lowastminus 1198821)119879

1198634]

4 Journal of Discrete Mathematics

= Re tr [1198771198671(1198601119881lowast1198611+ 1198621119882lowast1198631+ 1198602119881lowast1198612

+ 1198622119882lowast1198632+ 1198603119881lowast119867

1198613+ 1198623119882lowast119867

1198633

+ 1198604119881lowast119879

1198614+ 1198624119882lowast119879

1198634minus 119860111988111198611

minus 119862111988211198631minus 119860211988111198612minus 119862211988211198632

minus 1198603119881119867

11198613minus 1198623119882119867

11198633

minus 1198604119881119879

11198614minus 1198624119882119879

11198634)]

(10)

In view that 119881lowast and 119882lowast are solutions of matrix equation (1)

with this relation we have

Re tr [1198751198671

(119881lowastminus 1198811) + 119876119867

1(119882lowastminus 1198821)]

= Re tr [1198771198671(119864 minus 119860

111988111198611minus 119862111988211198631

minus 119860211988111198612

minus 119862211988211198632

minus 1198603119881119867

11198613

minus 1198623119882119867

11198633

minus 1198604119881119879

11198614minus 1198624119882119879

11198634)]

= Re tr [11987711986711198771] =

100381710038171003817100381711987711003817100381710038171003817

2

(11)

This implies that (8) holds for 119894 = 1Assume that (8) holds for = 119896 That is

Re tr [119875119867119896

(119881lowastminus 119881119896) + 119876119867

119896(119882lowastminus 119882119896)] =

10038171003817100381710038171198771198961003817100381710038171003817

2

(12)

Then we have to prove that the conclusion holds for 119894 = 119896+ 1it follows from Algorithm 3 that

Re tr [119875119867119896+1

(119881lowastminus 119881119896+1

) + 119876119867

119896+1(119882lowastminus 119882119896+1

)]

= Re

tr[

[

(119860119867

1119877119896+1

119861119867

1+ 119860119867

2119877119896+1

119861119867

2

+ 1198613119877119867

119896+11198603+ 1198614119877119879

119896+11198604

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2119875119896)

119867

(119881lowastminus 119881119896+1

)

+ (119862119867

1119877119896+1

119863119867

1+ 119862119867

2119877119896+1

119863119867

2

+ 1198633119877119867

119896+11198623+ 1198634119877119879

119896+11198624

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2119876119896)

119867

(119882lowastminus 119882119896+1

)]

]

= Retr[119877119867119896+1

1198601(119881lowastminus 119881119896+1

) 1198611

+ 119877119879

119896+11198602(119881lowastminus 119881119896+1

) 1198612

+ 119877119896+1

119861119867

3(119881lowastminus 119881119896+1

) 119860119867

3

+ 119877119896+1

1198614

119867

(119881lowastminus 119881119896+1

) 1198604

119867

+ 119877119867

119896+11198621(119882lowastminus 119882119896+1

)1198631

+ 119877119879

119896+11198622(119882lowastminus 119882119896+1

)1198632

+ 119877119896+1

119863119867

3(119882lowastminus 119882119896+1

) 119862119867

3

+ 119877119896+1

1198634

119867

(119882lowastminus 119882119896+1

) 1198624

119867

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2(119875119867

119896(119881lowastminus 119881119896+1

)

+ 119876119867

119896(119882lowastminus 119882119896+1

))]

(13)

From properties of trace and conjugate we get

Re tr [119875119867119896+1

(119881lowastminus 119881119896+1

) + 119876119867

119896+1(119882lowastminus 119882119896+1

)]

= Retr[119877119867119896+1

1198601(119881lowastminus 119881119896+1

) 1198611

+ 119877119879

119896+11198602(119881lowast minus 119881

119896+1) 1198612

+ 1198603(119881lowastminus 119881119896+1

)119867

1198613119877119867

119896+1

+ 1198604(119881lowastminus 119881119896+1

)119879

1198614119877119867

119896+1

+ 119877119867

119896+11198621(119882lowastminus 119882119896+1

)1198631

+ 119877119879

119896+11198622(119882lowast minus 119882

119896+1)1198632

+ 1198623(119882lowastminus 119882119896+1

)119867

1198633119877119867

119896+1

+ 1198624(119882lowastminus 119882119896+1

)119879

1198634119877119867

119896+1+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2

times (119875119867

119896(119881lowastminus 119881119896+1

)

+ 119876119867

119896(119882lowastminus 119882119896+1

))]

Journal of Discrete Mathematics 5

= Re tr [119877119867119896+1

1198601(119881lowastminus 119881119896+1

) 1198611

+ 119877119867

119896+11198602(119881lowast minus 119881

119896+1)1198612

+ 119877119867

119896+11198603(119881lowastminus 119881119896+1

)119867

1198613

+ 119877119867

119896+11198604(119881lowastminus 119881119896+1

)119879

1198614

+ 119877119867

119896+11198621(119882lowastminus 119882119896+1

)1198631

+ 119877119867

119896+11198622(119882lowast minus 119882

119896+1)1198632

+ 119877119867

119896+11198623(119882lowastminus 119882119896+1

)119867

1198633

+ 119877119867

119896+11198624(119882lowastminus 119882119896+1

)119879

1198634] +

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2

times Retr[119875119867119896

(119881lowastminus 119881119896minus

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2119875119896)

+ 119876119867

119896(119882lowastminus 119882119896

minus

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2119876119896)]

= Re tr [119877119867119896+1

(1198601119881lowast1198611+ 1198621119882lowast1198631

+ 1198602119881lowast1198612+ 1198622119882lowast1198632

+ 1198603119881lowast119867

1198613+ 1198623119882lowast119867

1198633

+ 1198604119881lowast119879

1198614+ 1198624119882lowast119879

1198634

minus 1198601119881119896+1

1198611minus 1198621119882119896+1

1198631minus 1198602119881119896+1

1198612

minus 1198622119882119896+1

1198632

minus 1198603119881119867

119896+11198613

minus 1198623119882119867

119896+11198633minus 1198604119881119879

119896+11198614

minus 1198624119882119879

119896+11198634)]

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2Retr[119875119867

119896(119881lowastminus 119881119896) + 119876

119867

119896(119882lowastminus 119882119896)

minus

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2(119875119867

119896119875119896+ 119876119867

119896119876119896)]

(14)

In view that 119881lowast and 119882lowast are solutions of matrix equation (1)

with relation (14) one has

Re tr [119875119867119896+1

(119881lowastminus 119881119896+1

) + 119876119867

119896+1(119882lowastminus 119882119896+1

)]

= Re tr [119877119867119896+1

(119864 minus 1198601119881119896+1

1198611minus 1198621119882119896+1

1198631

minus 1198602119881119896+1

1198612minus 1198622119882119896+1

1198632

minus 1198603119881119867

119896+11198613minus 1198623119882119867

119896+11198633

minus 1198604119881119879

119896+11198614minus 1198624119882119879

119896+11198634)]

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2[1003817100381710038171003817119877119896

1003817100381710038171003817

2

minus

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2(1003817100381710038171003817119875119896

1003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

)]

= Re tr [119877119867119896+1

119877119896+1

] =1003817100381710038171003817119877119896+1

1003817100381710038171003817

2

(15)

Then relation (8) holds by mathematical induction

Lemma 5 Suppose that the matrix equation (1) is consis-tent and the sequences 119877

119894 119875119894 and 119876

119894 are generated

by Algorithm 3 with any initial matrices 1198811 1198821 such that

119877119894

= 0 for all 119894 = 1 2 119896 and then

Re tr (119877119867119895119877119894) = 0

Re tr (119875119867119895119875119894+ 119876119867

119895119876119894) = 0 119894 119895 = 1 2 119896 119894 = 119895

(16)

Proof We apply mathematical inductionStep 1 We prove that

Re tr (119877119867119894+1

119877119894) = 0 (17)

Re tr (119875119867119894+1

119875119894+ 119876119867

119894+1119876119894) = 0 119894 = 1 2 119896 (18)

First from Algorithm 3 we have

119877119896+1

= 119864 minus 1198601119881119896+1

1198611minus 1198621119882119896+1

1198631minus 1198602119881119896+1

1198612

minus 1198622119882119896+1

1198632minus 1198603119881119867

119896+11198613

minus 1198623119882119867

119896+11198633minus 1198604119881119879

119896+11198614minus 1198624119882119879

119896+11198634

119877119896+1

= 119864 minus 1198601(119881119896+

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2119875119896)1198611

minus 1198621(119882119896+

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2119876119896)1198631

minus 1198602(119881119896+

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2119875119896)1198612

minus 1198622(119882119896+

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2119876119896)1198632

minus 1198603(119881119896+

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2119875119896)

119867

1198613

minus 1198623(119882119896+

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2119876119896)

119867

1198633

minus 1198604(119881119896+

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2119875119896)

119879

1198614

6 Journal of Discrete Mathematics

minus 1198624(119882119896+

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2119876119896)

119879

1198634

= 119864 minus 11986011198811198961198611minus 11986211198821198961198631minus 11986021198811198961198612minus 11986221198821198961198632

minus 1198603119881119867

1198961198613minus 1198623119882119867

1198961198633minus 1198604119881119879

1198961198614

minus 1198624119882119879

1198961198634minus

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

times (11986011198751198961198611+ 11986211198761198961198631+ 11986021198751198961198612

+ 11986221198761198961198632+ 1198603119875119867

1198961198613

+ 1198623119876119867

1198961198633+ 1198604119875119879

1198961198614+ 1198624119876119879

1198961198634)

119877119896+1

= 119877119896minus

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

times (11986011198751198961198611+ 11986211198761198961198631+ 11986021198751198961198612

+ 11986221198761198961198632+ 1198603119875119867

1198961198613

+ 1198623119876119867

1198961198633+ 1198604119875119879

1198961198614+ 1198624119876119879

1198961198634)

(19)

For 119894 = 1 it follows from (19) that

Re tr (11987711986721198771)

= Re

tr[

[

(1198771minus

100381710038171003817100381711987711003817100381710038171003817

2

100381710038171003817100381711987511003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2

times (119860111987511198611+ 119862111987611198631

+ 119860211987511198612+ 119862211987611198632

+ 1198603119875119867

11198613+ 1198623119876119867

11198633

+ 1198604119875119879

11198614+ 1198624119876119879

11198634))

119867

1198771]

]

= Retr (11987711986711198771) minus

100381710038171003817100381711987711003817100381710038171003817

2

100381710038171003817100381711987511003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2

times tr (1198751198671119860119867

11198771119861119867

1+ 119876119867

1119862119867

11198771119863119867

1

+ 1198751

119867

119860119867

21198771119861119867

2+ 1198761

119867

119862119867

21198771119863119867

2

+ 1198751119860119867

31198771119861119867

3+ 1198761119862119867

31198771119863119867

3

+ 1198751119860119867

41198771119861119867

4+ 1198761119862119867

41198771119863119867

4)

=10038171003817100381710038171198771

1003817100381710038171003817

2

minus

100381710038171003817100381711987711003817100381710038171003817

2

100381710038171003817100381711987511003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2

times Re tr [1198751198671119860119867

11198771119861119867

1+ 119876119867

1119862119867

11198771119863119867

1

+ 1198751

119867

119860119867

21198771119861119867

2+ 1198761

119867

119862119867

21198771119863119867

2

+ 1198613119877119867

11198603119875119867

1+ 1198633119877119867

11198623119876119867

1

+ 1198614119877119879

11198604119875119867

1+ 1198634119877119879

11198624119876119867

1]

=10038171003817100381710038171198771

1003817100381710038171003817

2

minus

100381710038171003817100381711987711003817100381710038171003817

2

100381710038171003817100381711987511003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2

times Re tr [1198751198671

(119860119867

11198771119861119867

1+ 119860119867

21198771119861119867

2

+ 1198613119877119867

11198603+ 1198614119877119879

11198604)

+ 119876119867

1(119862119867

11198771119863119867

1+ 119862119867

21198771119863119867

2

+ 1198633119877119867

11198623+ 1198634119877119879

11198624)]

=10038171003817100381710038171198771

1003817100381710038171003817

2

minus

100381710038171003817100381711987711003817100381710038171003817

2

100381710038171003817100381711987511003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2Re tr (119875119867

11198751+ 119876119867

11198761)

Re tr (11987711986721198771)

=10038171003817100381710038171198771

1003817100381710038171003817

2

minus

100381710038171003817100381711987711003817100381710038171003817

2

100381710038171003817100381711987511003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2(10038171003817100381710038171198751

1003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2

) = 0

(20)

This implies that (17) is satisfied for 119894 = 1From Algorithm 3 we have

Re tr (11987511986721198751+ 119876119867

21198761)

= Re

tr[

[

(119860119867

11198772119861119867

1+ 119860119867

21198772119861119867

2+ 1198613119877119867

21198603

+ 1198614119877119879

21198604+

100381710038171003817100381711987721003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

21198751)

119867

1198751

+ (119862119867

11198772119863119867

1+ 119862119867

21198772119863119867

2+ 1198633119877119867

21198623

+1198634119877119879

21198624+

100381710038171003817100381711987721003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

21198761)

119867

1198761]

]

Journal of Discrete Mathematics 7

= Retr[1198771198672119860111987511198611+ 1198772

119867

119860211987511198612+ 1198772119861119867

31198751119860119867

3

+ 11987721198614

119867

11987511198604

119867

+

100381710038171003817100381711987721003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

2119875119867

11198751+ 119877119867

2119862111987611198631

+ 1198772

119867

119862211987611198632+ 1198772119863119867

31198761119862119867

3

+ 11987721198634

119867

11987611198624

119867

+

100381710038171003817100381711987721003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

2119876119867

11198761]

= Retr[1198771198672119860111987511198611+ 1198772

119867

119860211987511198612+ 1198603119875119867

11198613119877119867

2

+ 1198604119875119879

11198614119877119867

2+ 119877119867

2119862111987611198631

+ 1198772

119867

119862211987611198632+ 1198623119876119867

11198633119877119867

2

+ 1198624119876119879

11198634119877119867

2+

100381710038171003817100381711987721003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

2(119875119867

11198751+ 119876119867

11198761)]

= Retr[1198771198672119860111987511198611+ 119877119867

2119860211987511198612+ 119877119867

21198603119875119867

11198613

+ 119877119867

21198604119875119879

11198614+ 119877119867

2119862111987611198631

+ 119877119867

2119862211987611198632+ 119877119867

21198623119876119867

11198633

+ 119877119867

21198624119876119879

11198634+

100381710038171003817100381711987721003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

2(119875119867

11198751+ 119876119867

11198761)]

= Retr[1198771198672(119860111987511198611+ 119862111987611198631+ 119860211987511198612

+ 119862211987611198632+ 1198603119875119867

11198613+ 1198623119876119867

11198633

+1198604119875119879

11198614+ 1198624119876119867

11198634) ]

+

100381710038171003817100381711987721003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

2(10038171003817100381710038171198751

1003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2

)

=

100381710038171003817100381711987511003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

2Re tr [119877119867

2(1198771minus 1198772)]

+

100381710038171003817100381711987721003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

2(10038171003817100381710038171198751

1003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2

)

= minus

100381710038171003817100381711987511003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

2(10038171003817100381710038171198772

1003817100381710038171003817

2

)

+

100381710038171003817100381711987721003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

2(10038171003817100381710038171198751

1003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2

) = 0

(21)This implies that (18) is satisfied for 119894 = 1

Assume that (17) and (18) hold for 119894 = 119896 minus 1 fromAlgorithm 3 we have

Re tr (119877119867119896+1

119877119896)

= Re

tr[

[

(119877119896minus

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

times (11986011198751198961198611+ 11986211198761198961198631+ 11986021198751198961198612

+ 11986221198761198961198632+ 1198603119875119867

1198961198613+ 1198623119876119867

1198961198633

+ 1198604119875119879

1198961198614+ 1198624119876119879

1198961198634))

119867

119877119896]

]

Re tr (119877119867119896+1

119877119896)

= Retr (119877119867119896119877119896) minus

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

times tr (119875119867119896119860119867

1119877119896119861119867

1+ 119876119867

119896119862119867

1119877119896119863119867

1

+ 119875119896

119867

119860119867

2119877119896119861119867

2+ 119876119896

119867

119862119867

2119877119896119863119867

2

+ 119875119896119860119867

3119877119896119861119867

3+ 119876119896119862119867

3119877119896119863119867

3

+ 119875119896119860119867

4119877119896119861119867

4+ 119876119896119862119867

4119877119896119863119867

4)

=1003817100381710038171003817119877119896

1003817100381710038171003817

2

minus

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

times Re tr (119875119867119896119860119867

1119877119896119861119867

1+ 119876119867

119896119862119867

1119877119896119863119867

1

+ 119875119896

119867

119860119867

2119877119896119861119867

2+ 119876119896

119867

119862119867

2119877119896119863119867

2

+ 1198613119877119867

1198961198603119875119867

119896+ 1198633119877119867

1198961198623119876119867

119896

+ 1198614119877119879

1198961198604119875119867

119896+ 1198634119877119879

1198961198624119876119867

119896)

=10038171003817100381710038171198771

1003817100381710038171003817

2

minus

100381710038171003817100381711987711003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

times Re tr [119875119867119896

(119860119867

1119877119896119861119867

1+ 119860119867

2119877119896119861119867

2

+ 1198613119877119867

1198961198603+ 1198614119877119879

1198961198604)

+ 119876119867

119896(119862119867

1119877119896119863119867

1+ 119862119867

2119877119896119863119867

2

+ 1198633119877119867

1198961198623+ 1198634119877119879

1198961198624)]

8 Journal of Discrete Mathematics

=1003817100381710038171003817119877119896

1003817100381710038171003817

2

minus

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

times Retr[119875119867119896

(119875119896minus

10038171003817100381710038171198771198961003817100381710038171003817

2

1003817100381710038171003817119877119896minus11003817100381710038171003817

2119875119896minus1

)

+ 119876119867

119896(119876119896minus

10038171003817100381710038171198771198961003817100381710038171003817

2

1003817100381710038171003817119877119896minus11003817100381710038171003817

2119876119896minus1

)]

=1003817100381710038171003817119877119896

1003817100381710038171003817

2

minus

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2(1003817100381710038171003817119875119896

1003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

) = 0

(22)

Thus (17) holds for 119894 = 119896Also from Algorithm 3 we have

Re tr (119875119867119896+1

119875119896+ 119876119867

119896+1119876119896)

= Re

tr [

[

(119860119867

1119877119896+1

119861119867

1+ 119860119867

2119877119896+1

119861119867

2

+ 1198613119877119867

119896+11198603+ 1198614119877119879

119896+11198604

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2119875119896)

119867

119875119896

+ (119862119867

1119877119896+1

119863119867

1+ 119862119867

2119877119896+1

119863119867

2

+ 1198633119877119867

119896+11198623+ 1198634119877119879

119896+11198624

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2119876119896)

119867

119876119896]

]

= Retr[119877119867119896+1

11986011198751198961198611+ 119877119896+1

119867

11986021198751198961198612

+ 119877119896+1

119861119867

3119875119896119860119867

3+ 119877119896+1

1198614

119867

1198751198961198604

119867

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2119875119867

119896119875119896+ 119877119867

119896+111986211198761198961198631

+ 119877119896+1

119867

11986221198761198961198632+ 119877119896+1

119863119867

3119876119896119862119867

3

+ 119877119896+1

1198634

119867

1198761198961198624

119867

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2119876119867

119896119876119896]

= Retr[119877119867119896+1

11986011198751198961198611+ 119877119896+1

119867

11986021198751198961198612

+ 1198603119875119867

1198961198613119877119867

119896+1+ 1198604119875119879

1198961198614119877119867

119896+1

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2119875119867

119896119875119896+ 119877119867

119896+111986211198761198961198631

+ 119877119896+1

119867

11986221198761198961198632+ 1198623119876119867

1198961198633119877119867

119896+1

+ 1198624119876119879

1198961198634119877119867

119896+1+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2119876119867

119896119876119896]

= Retr[119877119867119896+1

(11986011198751198961198611+ 11986211198761198961198631

+ 11986021198751198961198612+ 11986221198761198961198632+ 1198603119875119867

1198961198613

+1198623119876119867

1198961198633+ 1198604119875119879

1198961198614+ 1198624119876119879

1198961198634)

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2(119875119867

119896119875119896+ 119876119867

119896119876119896)]

=

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2Re tr (119877119867

119896+1(119877119896minus 119877119896+1

))

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2(1003817100381710038171003817119875119896

1003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

)

= minus

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2(1003817100381710038171003817119877119896+1

1003817100381710038171003817

2

)

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2(1003817100381710038171003817119875119896

1003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

) = 0

(23)

This implies that (17) and (18) hold for 119894 = 119896Then relations (17) and (18) holds bymathematical induc-

tionStep 2 We want to show that

Re (tr (119877119867119894+119897119877119894)) = 0

Re (tr (119875119867119894+119897119875119894+ 119876119867

119894+119897119876119894)) = 0

(24)

holds for 119897 ge 1 We will prove this conclusion by inductionThe case of 119897 = 1 has been proven in Step 1 Now we assumethat (24) holds for 119897 le 119904 119904 ge 1 The aim is to show that

Re (tr (119877119867119894+119904+1

119877119894)) = 0

Re (tr (119875119867119894+119904+1

119875119894+ 119876119867

119894+119904+1119876119894)) = 0

(25)

First we prove the following

Re (tr (119877119867119904+1

1198770)) = 0

Re (tr (119875119867119904+1

1198750+ 119876119867

119904+11198760)) = 0

(26)

Journal of Discrete Mathematics 9

By using Algorithm 3 from (19) and induction we have

Re tr (119877119867119904+1

1198770)

= Re

tr[

[

(119877119904minus

10038171003817100381710038171198771199041003817100381710038171003817

2

10038171003817100381710038171198751199041003817100381710038171003817

2

+1003817100381710038171003817119876119904

1003817100381710038171003817

2

times (11986011198751199041198611+ 11986211198761199041198631+ 11986021198751199041198612

+ 11986221198761199041198632+ 1198603119875119867

1199041198613+ 1198623119876119867

1199041198633

+ 1198604119875119879

1199041198614+ 1198624119876119879

1199041198634))

119867

1198770]

]

= Re tr (1198771198671199041198770) minus

10038171003817100381710038171198771199041003817100381710038171003817

2

10038171003817100381710038171198751199041003817100381710038171003817

2

+1003817100381710038171003817119876119904

1003817100381710038171003817

2

times tr (119875119867119904119860119867

11198770119861119867

1+ 119876119867

119904119862119867

11198770119863119867

1

+ 119875119904

119867

119860119867

21198770119861119867

2+ 119876119904

119867

119862119867

21198770119863119867

2

+ 119875119904119860119867

31198770119861119867

3+ 119876119904119862119867

31198770119863119867

3

+ 119875119904119860119867

41198770119861119867

4+ 119876119904119862119867

41198770119863119867

4)

Re tr (119877119867119904+1

1198770)

= Re tr (1198771198671199041198770) minus

10038171003817100381710038171198771199041003817100381710038171003817

2

10038171003817100381710038171198751199041003817100381710038171003817

2

+1003817100381710038171003817119876119904

1003817100381710038171003817

2

times Re tr (119875119867119904119860119867

11198770119861119867

1+ 119876119867

119904119862119867

11198770119863119867

1

+ 119875119904

119867

119860119867

21198770119861119867

2+ 119876119904

119867

119862119867

21198770119863119867

2

+ 1198613119877119867

01198603119875119867

119904+ 1198633119877119867

01198623119876119867

119904

+ 1198614119877119879

01198604119875119867

119904+ 1198634119877119879

01198624119876119867

119904)

= minus

10038171003817100381710038171198771199041003817100381710038171003817

2

10038171003817100381710038171198751199041003817100381710038171003817

2

+1003817100381710038171003817119876119904

1003817100381710038171003817

2

times Re tr (119875119867119904

(119860119867

11198770119861119867

1+ 119860119867

21198770119861119867

2

+ 1198613119877119867

01198603+ 1198614119877119879

01198604)

+ 119876119867

119904(119862119867

11198770119863119867

1+ 119862119867

21198770119863119867

2

+1198633119877119867

01198623+ 1198634119877119879

01198624))

= minus

10038171003817100381710038171198771199041003817100381710038171003817

2

10038171003817100381710038171198751199041003817100381710038171003817

2

+1003817100381710038171003817119876119904

1003817100381710038171003817

2Re tr (119875119867

1199041198750+ 119876119867

1199041198760) = 0

Re tr (119875119867119904+1

1198750+ 119876119867

119904+11198760)

= Retr[(119860119867

1119877119904+1

119861119867

1+ 119860119867

2119877119904+1

119861119867

2

+ 1198613119877119867

119904+11198603+ 1198614119877119879

119904+11198604

+

1003817100381710038171003817119877119904+11003817100381710038171003817

2

10038171003817100381710038171198771199041003817100381710038171003817

2119875119904)

119867

1198750

+ (119862119867

1119877119904+1

119863119867

1+ 119862119867

2119877119904+1

119863119867

2

+ 1198633119877119867

119904+11198623+ 1198634119877119879

119896+11198624

+

1003817100381710038171003817119877119904+11003817100381710038171003817

2

10038171003817100381710038171198771199041003817100381710038171003817

2119876119904)

119867

1198760]

]

= Retr[119877119867119904+1

119860111987501198611+ 119877119904+1

119867

119860211987501198612

+ 119877119904+1

119861119867

31198750119860119867

3+ 119877119904+1

1198614

119867

11987501198604

119867

+

1003817100381710038171003817119877119904+11003817100381710038171003817

2

10038171003817100381710038171198771199041003817100381710038171003817

2119875119867

1199041198750+ 119877119867

119904+1119862111987601198631

+ 119877119904+1

119867

119862211987601198632+ 119877119904+1

119863119867

31198760119862119867

3

+ 119877119904+1

1198634

119867

11987601198624

119867

+

1003817100381710038171003817119877119904+11003817100381710038171003817

2

10038171003817100381710038171198771199041003817100381710038171003817

2119876119867

1199041198760]

= Retr[119877119867119904+1

119860111987501198611+ 119877119904+1

119867

119860211987501198612

+ 1198603119875119867

01198613119877119867

119904+1+ 1198604119875119879

01198614119877119867

119904+1

+

1003817100381710038171003817119877119904+11003817100381710038171003817

2

10038171003817100381710038171198771199041003817100381710038171003817

2119875119867

1199041198750+ 119877119867

119904+1119862111987601198631

+ 119877119904+1

119867

119862211987601198632+ 1198623119876119867

01198633119877119867

119904+1

+ 1198624119876119879

01198634119877119867

119904+1+

1003817100381710038171003817119877119904+11003817100381710038171003817

2

10038171003817100381710038171198771199041003817100381710038171003817

2119876119867

1199041198760]

= Retr[119877119867119904+1

(119860111987501198611+ 119862111987601198631+ 119860211987501198612

+ 119862211987601198632+ 1198603119875119867

01198613

+ 1198623119876119867

01198633+ 1198604119875119879

01198614

+ 1198624119876119879

01198634)

+

1003817100381710038171003817119877119904+11003817100381710038171003817

2

10038171003817100381710038171198771199041003817100381710038171003817

2(119875119867

1199041198750+ 119876119867

1199041198760)]

=

100381710038171003817100381711987501003817100381710038171003817

2

+10038171003817100381710038171198760

1003817100381710038171003817

2

100381710038171003817100381711987701003817100381710038171003817

2Re tr (119877119867

119904+1(1198770minus 1198771))= 0

(27)

Then (26) is holds

10 Journal of Discrete Mathematics

From Algorithm 3 we have

Re tr (119875119867119894+119904+1

119875119894+ 119876119867

119894+119904+1119876119894)

= Retr[(119860119867

1119877119894+119904+1

119861119867

1+ 119860119867

2119877119894+119904+1

119861119867

2

+ 1198613119877119867

119894+119904+11198603+ 1198614119877119879

119894+119904+11198604

+

1003817100381710038171003817119877119894+119904+11003817100381710038171003817

2

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2119875119894+119904

)

119867

119875119894

+ (119862119867

1119877119894+119904+1

119863119867

1+ 119862119867

2119877119894+119904+1

119863119867

2

+ 1198633119877119867

119894+119904+11198623+ 1198634119877119879

119894+119904+11198624

+

1003817100381710038171003817119877119894+119904+11003817100381710038171003817

2

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2119876119894+119904

)

119867

119876119894]

]

= Retr[119877119867119894+119904+1

11986011198751198941198611+ 119877119894+119904+1

119867

11986021198751198941198612

+ 119877119894+119904+1

119861119867

3119875119894119860119867

3+ 119877119894+119904+1

1198614

119867

1198751198941198604

119867

+

1003817100381710038171003817119877119894+119904+11003817100381710038171003817

2

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2119875119867

119894+119904119875119894+ 119877119867

119894+119904+111986211198761198941198631

+ 119877119894+119904+1

119867

11986221198761198941198632+ 119877119894+119904+1

119863119867

3119876119894119862119867

3

+ 119877119894+119904+1

1198634

119867

1198761198941198624

119867

+

1003817100381710038171003817119877119894+119904+11003817100381710038171003817

2

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2119876119867

119894+119904119876119894]

= Retr[119877119867119894+119904+1

11986011198751198941198611+ 119877119894+119904+1

119867

11986021198751198941198612

+ 1198603119875119867

1198941198613119877119867

119894+119904+1+ 1198604119875119879

1198941198614119877119867

119894+119904+1

+

1003817100381710038171003817119877119894+119904+11003817100381710038171003817

2

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2119875119867

119894+119904119875119894+ 119877119867

119894+119904+111986211198761198941198631

+ 119877119894+119904+1

119867

11986221198761198941198632+ 1198623119876119867

1198941198633119877119867

119894+119904+1

+ 1198624119876119879

1198961198634119877119867

119894+119904+1+

1003817100381710038171003817119877119894+119904+11003817100381710038171003817

2

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2119876119867

119894+119904119876119894]

= Retr[119877119867119894+119904+1

(11986011198751198941198611+ 11986211198761198941198631

+ 11986021198751198941198612+ 11986221198761198941198632

+ 1198603119875119867

1198941198613+ 1198623119875119867

1198941198633

+ 1198604119875119879

1198941198614+ 1198624119876119879

1198941198634)

+

1003817100381710038171003817119877119894+119904+11003817100381710038171003817

2

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2(119875119867

119894+119904119875119894+ 119876119867

119894+119904119876119894)]

=

10038171003817100381710038171198751198941003817100381710038171003817

2

+1003817100381710038171003817119876119894

1003817100381710038171003817

2

10038171003817100381710038171198771198941003817100381710038171003817

2Re tr (119877119867

119894+119904+1(119877119894minus 119877119894+1

))

=

10038171003817100381710038171198751198941003817100381710038171003817

2

+1003817100381710038171003817119876119894

1003817100381710038171003817

2

10038171003817100381710038171198771198941003817100381710038171003817

2Re tr (119877119867

119894+119904+1119877119894)

(28)

Also from (19) we have

Re tr (119877119867119894+119904+1

119877119894)

= Re

tr[

[

(119877119894+119904

minus

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2

1003817100381710038171003817119875119894+1199041003817100381710038171003817

2

+1003817100381710038171003817119876119894+119904

1003817100381710038171003817

2

times (1198601119875119894+119904

1198611+ 1198621119876119894+119904

1198631

+ 1198602119875119894+119904

1198612+ 1198622119876119894+119904

1198632

+ 1198603119875119867

119894+1199041198613+ 1198623119876119867

119894+1199041198633

+ 1198604119875119879

119894+1199041198614+ 1198624119876119879

119894+1199041198634))

119867

119877119894]

]

= Retr (119877119867119894+119904

119877119894) minus

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2

1003817100381710038171003817119875119894+1199041003817100381710038171003817

2

+1003817100381710038171003817119876119894+119904

1003817100381710038171003817

2

times tr (119875119867119894+119904

119860119867

1119877119894119861119867

1+ 119876119867

119894+119904119862119867

1119877119894119863119867

1

+ 119875119894+119904

119867

119860119867

2119877119894119861119867

2+ 119876119894+119904

119867

119862119867

2119877119894119863119867

2

+ 119875119894+119904

119860119867

3119877119894119861119867

3+ 119876119894+119904

119862119867

3119877119894119863119867

3

+ 119875119894+119904

119860119867

4119877119894119861119867

4+ 119876119894+119904

119862119867

4119877119894119863119867

4)

Re tr (119877119867119894+119904+1

119877119894)

= minus

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2

1003817100381710038171003817119875119894+1199041003817100381710038171003817

2

+1003817100381710038171003817119876119894+119904

1003817100381710038171003817

2

times Retr(119875119867119894+119904

119860119867

1119877119894119861119867

1+ 119876119867

119894+119904119862119867

1119877119894119863119867

1

+ 119875119894+119904

119867

119860119867

2119877119894119861119867

2+ 119876119894+119904

119867

119862119867

2119877119894119863119867

2

+ 1198613119877119867

1198941198603119875119867

119894+119904+ 1198633119877119867

1198941198623119876119867

119894+119904

+ 1198614119877119879

1198941198604119875119867

119894+119904+ 1198634119877119879

1198941198624119876119867

119894+119904)

Journal of Discrete Mathematics 11

= minus

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2

1003817100381710038171003817119875119894+1199041003817100381710038171003817

2

+1003817100381710038171003817119876119894+119904

1003817100381710038171003817

2

times Re tr [119875119867119894+119904

(119860119867

1119877119894119861119867

1+ 119860119867

2119877119894119861119867

2

+ 1198613119877119867

1198941198603+ 1198614119877119879

1198941198604)

+ 119876119867

119894+119904(119862119867

1119877119894119863119867

1+ 119862119867

2119877119894119863119867

2

+ 1198633119877119867

1198941198623+ 1198634119877119879

1198941198624)]

= minus

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2

1003817100381710038171003817119875119894+1199041003817100381710038171003817

2

+1003817100381710038171003817119876119894+119904

1003817100381710038171003817

2

times Re tr[119875119867119894+119904

(119875119894minus

10038171003817100381710038171198771198941003817100381710038171003817

2

1003817100381710038171003817119877119894minus11003817100381710038171003817

2119875119894minus1

)

+ 119876119867

119894+119904(119876119894minus

10038171003817100381710038171198771198941003817100381710038171003817

2

1003817100381710038171003817119877119894minus11003817100381710038171003817

2119876119894minus1

)]

=

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2

1003817100381710038171003817119875119894+1199041003817100381710038171003817

2

+1003817100381710038171003817119876119894+119904

1003817100381710038171003817

2

10038171003817100381710038171198771198941003817100381710038171003817

2

1003817100381710038171003817119877119894minus11003817100381710038171003817

2

times Re tr (119875119867119894+119904

119875119894minus1

+ 119876119867

119894+119904119876119894minus1

)

(29)

Repeating (28) and (29) one can easily obtain for certain 120572

and 120573

tr (119875119867119894+119904+1

119875119894+ 119876119867

119894+119904+1119876119894) = 120572 [tr (119875119867

119904+11198751+ 119876119867

119904+11198761)]

tr (119877119867119894+119904+1

119877119894) = 120573 [tr (119877119867

119904+11198771)]

(30)

Combining these two relations with (26) implies that (24)holds for 119897 = 119904 + 1 From Steps 1 and 2 the conclusion holdsby the principle of inductionWith the above two lemmas wehave the following theorem

Theorem 6 (see [32]) If the matrix equation (1) is consistentthen a solution can be obtained within finite iteration steps byusing Algorithm 3 for any initial matrices 119881

11198821

4 Numerical Example

In this section we present numerical example to illustrate theapplication of our proposed methods

Example 7 In this example we illustrate our theoreticalresults of Algorithm 3 for solving the system of matrixequation

11986011198811198611+ 11986211198821198631+ 11986021198811198612+ 11986221198821198632+ 11986031198811198671198613

+ 11986231198821198671198633+ 11986041198811198791198614+ 11986241198821198791198634= 119864

(31)

Because of the influence of the error of calculation theresidual 119877(119896) is usually unequal to zero in this process ofthe iteration We regard the matrix 119877(119896) as a zero matrix if119877(119896) lt 10

minus10Given

1198601= [

2 + 3119894 minus119894 1 + 119894

5 1 + 2119894 minus3]

1198602= [

2 + 3119894 minus119894 1 + 119894

5 1 + 2119894 minus3]

1198603= [

0 2 minus 119894 119894

minus1 + 3119894 2 0] 119860

4=[

0 1 minus 3119894 1 + 119894

0 4 + 119894 minus3119894]

1198621= [

1 + 2119894 3 minus 119894 4

minus119894 2119894 minus3] 119862

2=[

3 + 2119894 0 1 + 119894

0 4119894 1 minus 2119894]

1198623= [

1 minus 3119894 2119894 minus3119894

1 2 + 3119894 4119894] 119862

4=[

1 minus 2119894 0 2

3 minus 119894 1 + 119894 minus1]

1198611= [

[

4 + 119894 minus119894

0 1 minus 119894

4119894 2 + 2119894

]

]

1198612= [

[

0 119894

1 + 119894 0

minus1 minus 119894 3119894

]

]

1198613= [

[

0 1

minus3119894 4 + 119894

5 1 + 2119894

]

]

1198614=[

[

3 + 119894 minus1 minus 119894

0 2 minus 119894

minus1 + 119894 2

]

]

1198631= [

[

0 0

1 minus 3119894 minus119894

2119894 minus3119894

]

]

1198632= [

[

0 119894

1 + 119894 0

minus1 minus 119894 3119894

]

]

1198633= [

[

0 1

minus3119894 4 + 119894

5 1 + 2119894

]

]

1198634= [

[

3119894 minus2 + 119894

0 119894

minus2119894 minus4119894

]

]

119864 = [42 + 55119894 115 + 25119894

minus38 minus 119894 132 + 44119894]

(32)

Taking 1198811= [0 0

0 0

0 0

] and 1198821= [0 0

0 0] we apply Algorithm 3

to compute 119881119896119882119896

And iterating 42 steps we get

119881

=[

[

00126 + 18415119894 00827 + 06381119894 11221 minus 09428119894

minus06903 + 10185119894 18818 + 10203119894 09208 + 04569119894

05344 minus 04909119894 09280 + 07169119894 04872 minus 02734119894

]

]

119882

=[

[

04218 minus 09710119894 01763 + 05183119894 11331 minus 00432119894

06273 minus 01216119894 minus03902 + 04313119894 06240 + 09828119894

minus07011 minus 03418119894 03695 + 17627119894 minus03032 + 07073119894

]

]

(33)

which satisfy the matrix equation

11986011198811198611+ 11986211198821198631+ 11986021198811198612+ 11986221198821198632+ 11986031198811198671198613

+ 11986231198821198671198633+ 11986041198811198791198614+ 11986241198821198791198634= 119864

(34)

12 Journal of Discrete Mathematics

log10 rk

4

2

0

minus2

minus4

minus6

minus8

minus100 5 10 15 20 25 30 35 40 45

k (iteration number)

Figure 1The relation between the number of iterations and residualfor the example

With the corresponding residual100381710038171003817100381711987742

1003817100381710038171003817

=10038171003817100381710038171003817119864 minus 119860

1119881421198611minus 1198621119882421198631minus 1198602119881421198612minus 1198622119882421198632

minus 1198603119881119867

421198613minus 1198623119882119867

421198633minus 1198604119881119879

421198614minus 1198624119882119879

421198634

10038171003817100381710038171003817

= 66115 times 10minus11

(35)

5 Conclusions

The above Figure 1 shows the convergence curve for theresidual function 119877(119896) In this paper an iterative algorithmconstructed to solve a complex matrix equation with conju-gate and transpose of two unknowns of the form 119860

11198811198611+

11986211198821198631+11986021198811198612+11986221198821198632+11986031198811198671198613+11986231198821198671198633+11986041198811198791198614+

11986241198821198791198634

= 119864 is presented We proved that the iterativealgorithms always converge to the solution for any initialmatrices We stated and proved some lemmas and theoremswhere the solutions are obtained The proposed methodis illustrated by numerical example where the obtainednumerical results show that our technique is very neat andefficient

References

[1] F Ding P X Liu and J Ding ldquoIterative solutions of thegeneralized Sylvester matrix equations by using the hierarchicalidentification principlerdquo Applied Mathematics and Computa-tion vol 197 no 1 pp 41ndash50 2008

[2] F Ding and T Chen ldquoGradient based iterative algorithmsfor solving a class of matrix equationsrdquo IEEE Transactions onAutomatic Control vol 50 no 8 pp 1216ndash1221 2005

[3] FDing andTChen ldquoHierarchical gradient-based identificationof multivariable discrete-time systemsrdquo Automatica vol 41 no2 pp 315ndash325 2005

[4] F Ding and T Chen ldquoIterative least-squares solutions of cou-pled Sylvester matrix equationsrdquo Systems and Control Lettersvol 54 no 2 pp 95ndash107 2005

[5] F Ding and T Chen ldquoOn iterative solutions of general coupledmatrix equationsrdquo SIAM Journal on Control and Optimizationvol 44 no 6 pp 2269ndash2284 2006

[6] F Ding and T Chen ldquoHierarchical least squares identificationmethods for multivariable systemsrdquo IEEE Transactions onAutomatic Control vol 50 no 3 pp 397ndash402 2005

[7] F Ding and T Chen ldquoPerformance analysis ofmulti-innovationgradient type identification methodsrdquo Automatica vol 43 no1 pp 1ndash14 2007

[8] AGWuX ZengG RDuan andW JWu ldquoIterative solutionsto the extended Sylvester-conjugate matrix equationsrdquo AppliedMathematics and Computation vol 217 no 1 pp 130ndash142 2010

[9] B Zhou G R Duan and Z Y Li ldquoGradient based iterativealgorithm for solving coupled matrix equationsrdquo Systems andControl Letters vol 58 no 5 pp 327ndash333 2009

[10] J H Bevis F J Hall and R E Hartwing ldquoConsimilarity andthe matrix equation119860119883minus119883119861 = 119862rdquo in Current Trends inMatrixTheory pp 51ndash64 North-Holland New York NY USA 1987

[11] R A Horn and C R Johnson Matrix Analysis CambridgeUniversity Press Cambridge UK 1990

[12] H Liping ldquoConsimilarity of quaternion matrices and complexmatricesrdquo Linear Algebra and Its Applications vol 331 no 1ndash3pp 21ndash30 2001

[13] T Jiang X Cheng and L Chen ldquoAn algebraic relation betweenconsimilarity and similarity of complex matrices and its appli-cationsrdquo Journal of Physics A vol 39 no 29 pp 9215ndash92222006

[14] J H Bevis F J Hall and R E Hartwig ldquoThe matrix equation119860119883 minus 119883119861 = 119862 and its special casesrdquo SIAM Journal on MatrixAnalysis and Applications vol 9 no 3 pp 348ndash359 1988

[15] A G Wu G R Duan and H H Yu ldquoOn solutions of thematrix equations 119883119865 minus 119860119883 = 119862 and 119883119865 minus 119860119883 = 119862rdquo AppliedMathematics and Computation vol 183 no 2 pp 932ndash9412006

[16] A Navarra P L Odell and D M Young ldquoRepresentationof the general common solution to the matrix equations11986011198831198611= 1198621 11986021198831198612= 1198622with applicationsrdquo Computers and

Mathematics with Applications vol 41 no 7-8 pp 929ndash9352001

[17] J W Van der Woude ldquoOn the existence of a common solutionX to the matrix equations 119860

119894119883119861119895= 119862 (i j) E Irdquo Linear Algebra

and Its Applications vol 375 no 1ndash3 pp 135ndash145 2003[18] P Bhimasankaram ldquoCommon solutions to the linear matrix

equations 119860119883 = 119862119883119861 = 119863 and EXF = Grdquo Sankhya Series Avol 38 pp 404ndash409 1976

[19] S Kumar Mitra ldquoThe matrix equations 119860119883 = 119862119883119861 = 119863rdquoLinear Algebra and Its Applications vol 59 pp 171ndash181 1984

[20] S K Mitra ldquoA pair of simultaneous linear matrix equations11986011198831198611

= 1198621 11986021198831198612

= 1198622and a matrix programming

problemrdquo Linear Algebra and Its Applications vol 131 pp 107ndash123 1990

[21] M A Ramadan M A Abdel Naby and A M E BayoumildquoOn the explicit solutions of forms of the Sylvester and theYakubovich matrix equationsrdquo Mathematical and ComputerModelling vol 50 no 9-10 pp 1400ndash1408 2009

[22] K W E Chu ldquoSingular value and generalized singular valuedecompositions and the solution of linear matrix equationsrdquoLinear Algebra and Its Applications vol 88-89 pp 83ndash98 1987

Journal of Discrete Mathematics 13

[23] Y X Yuan ldquoThe optimal solution of linear matrix equationby matrix decompositions Mathrdquo Numerica Sinica vol 24 pp165ndash176 2002

[24] A P Liao and Y Lei ldquoLeast-squares solution with the mini-mum-norm for the matrix equation (119860119883119861 119866119883119867) = (119862119863)rdquoComputers and Mathematics with Applications vol 50 no 3-4pp 539ndash549 2005

[25] Y H Liu ldquoRanks of least squares solutions of the matrix equa-tion 119860119883119861 = 119862rdquo Computers and Mathematics with Applicationsvol 55 no 6 pp 1270ndash1278 2008

[26] X Sheng and G Chen ldquoA finite iterative method for solving apair of linear matrix equations (119860119883119861 119862119883119863) = (119864 119865)rdquo AppliedMathematics and Computation vol 189 no 2 pp 1350ndash13582007

[27] A G Wu W Liu and G R Duan ldquoOn the conjugate productof complex polynomial matricesrdquo Mathematical and ComputerModelling vol 53 no 9-10 pp 2031ndash2043 2011

[28] A GWu G FengW Liu and G R Duan ldquoThe complete solu-tion to the Sylvester-polynomial-conjugate matrix equationsrdquoMathematical and Computer Modelling vol 53 no 9-10 pp2044ndash2056 2011

[29] A G Wu B Li Y Zhang and G R Duan ldquoFinite iterativesolutions to coupled Sylvester-conjugate matrix equationsrdquoApplied Mathematical Modelling vol 35 no 3 pp 1065ndash10802011

[30] AGWu L Lv andGRDuan ldquoIterative algorithms for solvinga class of complex conjugate and transpose matrix equationsrdquoApplied Mathematics and Computation vol 217 no 21 pp8343ndash8353 2011

[31] X ZhangMatrix Analysis and Application TsinghuaUniversityPress Beijing China 2004

[32] A G Wu L Lv and M Z Hou ldquoFinite iterative algorithms forextended Sylvester-conjugate matrix equationsrdquo Mathematicaland Computer Modelling vol 54 pp 2363ndash2384 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Journal of Discrete Mathematics

studied a representation of the general common solution ofthe matrix equations 119860

11198831198611

= 1198621and 119860

21198831198612

= 1198622[16]

Van derWoude obtained the existence of a common solution119883 for thematrix equations119860

119894119883119861119895= 119862119894119895[17] Bhimasankaram

considered the linear matrix equations 119860119883 = 119861 119862119883 = 119863and 119864119883119865 = 119866 [18] Mitra has provided conditions for theexistence of a solution and a representation of the generalcommon solution of thematrix equations119860119883 = 119862 and 119883119861 =

119863 and the matrix equations 11986011198831198611= 1198621and 119860

21198831198612= 1198622

[19 20] Ramadan et al [21] introduced a complete generaland explicit solution to the Yakubovich matrix equation 119881 minus

119860119881119865 = 119861119882 and the matrix equation (119860119883119861 119866119883119867) = (119862119863)

has some important results that have been developed In [22]necessary and sufficient conditions for its solvability and theexpression of the solution were derived by means of gener-alized inverse Moreover in [22] the least-squares solutionwas also obtained by using the generalized singular valuedecomposition While in [23] when this matrix equation isconsistent the minimum-norm solution was given by the useof the canonical correlation decomposition In [24] basedon the projection theorem in Hilbert space an analyticalexpression of the least-squares solution was given for thematrix equation (119860119883119861 119866119883119867) = (119862119863) by making use of thegeneralized singular value decomposition and the canonicalcorrelation decomposition In [25] by using the matrix rankmethod a necessary and sufficient condition was derivedfor the matrix equations 119860119883

1119861 = 119862 and 119866119883

2119867 = 119863 to

have a common least square solution In the aforementionedmethods the coefficient matrices of the considered equationsare required to be firstly transformed into some canonicalforms Recently an iterative algorithm has presented in [26]to solve thematrix equation (119860119883119861 119866119883119867) = (119862119863) Differentfrom the above mentioned methods this algorithm can beimplemented by initial coefficient matrices and can provide asolution within finite iteration steps for any initial values

Very recently in [27] a new operator of conjugate productfor complex polynomialmatrices is proposed It is shown thatan arbitrary complex polynomial matrix can be convertedinto the so-called Smith normal form by elementary transfor-mations in the framework of conjugate product Meanwhilethe conjugate product and the Sylvester-conjugate sum arealso proposed in [28] Based on the important properties ofthe above new operators a unified approach to solve a generalclass of Sylvester-polynomial-conjugate matrix equations isgiven The complete solution of the Sylvester-polynomial-conjugate matrix equation is obtained In [29] by using a realinner product in complex matrix spaces a solution can beobtained within finite iterative steps for any initial values inthe absence of round-off errors In [30] iterative solutions toa class of complex matrix equations are given by applying thehierarchical identification principle

This paper is organized as follows First in Section 2we introduce some notations a lemma and a theoremthat will be needed to develop this work In Section 3 wepropose iterativemethods to obtain numerical solution to thecomplexmatrix equationwith conjugate and transpose of twounknowns of the form119860

11198811198611+11986211198821198631+11986021198811198612+11986221198821198632+

11986031198811198671198613+11986231198821198671198633+11986041198811198791198614+11986241198821198791198634= 119864 using iterative

method In Section 4 numerical example is given to explorethe simplicity and the neatness of the presented methods

2 Preliminaries

The following notations definitions lemmas and theo-rems will be used to develop the proposed work We use119860119879 119860 119860

119867 tr(119860) and 119860 to denote the transpose conjugateconjugate transpose the trace and the Frobenius norm of amatrix119860 respectively We denote the set of all119898times119899 complexmatrices byC119898times119899 and Re(119886) denote the real part of number 119886

Definition 1 (inner product [31]) A real inner product spaceis a vector space119881 over the real fieldR together with an innerproduct That is with a map

⟨sdot sdot⟩ 119881 times 119881 997888rarr R (2)

Satisfying the following three axioms for all vectors 119909 119910 119911 isin

119881 and all scalars 119886 isin R

(1) symmetry ⟨119909 119910⟩ = ⟨119910 119909⟩

(2) linearity in the first argument

⟨119886119909 119910⟩ = 119886 ⟨119909 119910⟩

⟨119909 + 119910 119911⟩ = ⟨119909 119911⟩ + ⟨119910 119911⟩

(3)

(3) positive definiteness ⟨119909 119909⟩ gt 0 for all 119909 = 0

two vectors 119906 119907 isin 119881 are said to be orthogonal if ⟨119906 119907⟩ =

0The following theorem defines a real inner product on

space C119898times119899 over the field R

Theorem 2 (see [32]) In the space C119898times119899 over the field R aninner product can be defined as

⟨119860 119861⟩ = Re [tr (119860119867119861)] (4)

3 The Main Result

In this section we propose an iterative solution to a com-plex matrix equation with conjugate and transpose of twounknowns

11986011198811198611+ 11986211198821198631+ 11986021198811198612+ 11986221198821198632+ 11986031198811198671198613

+ 11986231198821198671198633+ 11986041198811198791198614+ 11986241198821198791198634= 119864

(5)

defined in (1) where 1198601 1198602 1198621 1198622

isin C119898times119903 1198611 1198612 1198631

1198632isin C119904times119899 119860

3 1198604 1198623 1198624isin C119898times119904 119864 isin C119898times119899 and 119861

3 1198614

1198633 1198634isin C119903times119899 are given matrices while 119881 119882 isin C119903times119904 are

matrices to be determinedThe following finite iterative algorithm is presented to

solve it

Algorithm 3 (1) Input1198601119860211986211198622 1198611 11986121198631119863211986031198604

1198623 1198624 1198613 119861411986331198634 119864

(2) Chosen arbitrary matrices 11988111198821isin C119903times119904

Journal of Discrete Mathematics 3

(3) Set

1198771= 119864 minus 119860

111988111198611minus 119862111988211198631minus 119860211988111198612

minus 119862211988211198632minus 1198603119881119867

11198613minus 1198623119882119867

11198633

minus 1198604119881119879

11198614minus 1198624119882119879

11198634

1198751= 119860119867

11198771119861119867

1+ 119860119867

21198771119861119867

2+ 1198613119877119867

11198603+ 1198614119877119879

11198604

1198761= 119862119867

11198771119863119867

1+ 119862119867

21198771119863119867

2+ 1198633119877119867

11198623+ 1198634119877119879

11198624

119896 = 1

(6)

(4) If 119877119896= 0 then stop else go to Step 5

(5) Set

119881119896+1

= 119881119896+

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2119875119896

119882119896+1

= 119882119896+

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2119876119896

119877119896+1

= 119864 minus 1198601119881119896+1

1198611minus 1198621119882119896+1

1198631minus 1198602119881119896+1

1198612

minus 1198622119882119896+1

1198632minus 1198603119881119867

119896+11198613minus 1198623119882119867

119896+11198633

minus 1198604119881119879

119896+11198614minus 1198624119882119879

119896+11198634

119875119896+1

= 119860119867

1119877119896+1

119861119867

1+ 119860119867

2119877119896+1

119861119867

2+ 1198613119877119867

119896+11198603

+ 1198614119877119879

119896+11198604+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2119875119896

119876119896+1

= 119862119867

1119877119896+1

119863119867

1+ 119862119867

2119877119896+1

119863119867

2

+ 1198633119877119867

119896+11198623+ 1198634119877119879

119896+11198624+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2119876119896

(7)

(6) If 119877119896+1

= 0 then stop else let 119896 = 119896 + 1 go to Step 5To prove the convergence property of Algorithm 3 we

first establish the following basic properties

Lemma 4 Suppose that the matrix equation (1) is consistentand 119881

lowast 119882lowast are arbitrary solutions of (1) Then for any initialmatrices 119881

1and119882

1 we have

Re tr [119875119867119894

(119881lowastminus 119881119894) + 119876119867

119894(119882lowastminus 119882119894)] =

10038171003817100381710038171198771198941003817100381710038171003817

2

(8)

where the sequence 119881119894 119875119894 119882119894 119876119894 and 119877

119894 are gener-

ated by Algorithm 3 for 119894 = 1 2

Proof We applymathematical induction to prove the conclu-sion

For 119894 = 1 from Algorithm 3 we have

Re tr [1198751198671

(119881lowastminus 1198811) + 119876119867

1(119882lowastminus 1198821)]

= Re tr [(11986011986711198771119861119867

1+ 119860119867

21198771119861119867

2

+ 1198613119877119867

11198603+ 1198614119877119879

11198604119861119867

2)

119867

(119881lowastminus 1198811)

+ (119862119867

11198771119863119867

1+ 119862119867

21198771119863119867

2

+1198633119877119867

11198623+ 1198634119877119879

11198624119863119867

2)

119867

times (119882lowastminus 1198821) ]

= Re tr [11987711986711198601(119881lowastminus 1198811) 1198611+ 1198771

119867

1198602(119881lowastminus 1198811) 1198612

+ 1198771119861119867

3(119881lowastminus 1198811) 119860119867

3+ 1198771119861119867

4(119881lowastminus 1198811) 119860119867

4

+ 119877119867

11198621(119882lowastminus 1198821)1198631+ 1198771

119867

1198622(119882lowastminus 1198821)1198632

+ 1198771119863119867

3(119882lowastminus 1198821) 119862119867

3

+ 1198771119863119867

4(119882lowastminus 1198821) 119862119867

4]

(9)

From properties of trace and conjugate

Re tr [1198751198671

(119881lowastminus 1198811) + 119876119867

1(119882lowastminus 1198821)]

= Retr [11987711986711198601(119881lowastminus 1198811) 1198611+ 1198771

119867

1198602(119881lowast minus 119881

1) 1198612

+ 1198603(119881lowastminus 1198811)119867

1198613119877119867

1+ 1198604(119881lowastminus 1198811)119879

1198614119877119867

1

+ 119877119867

11198621(119882lowastminus 1198821)1198631

+ 1198771

119867

1198622(119882lowast minus 119882

1)1198632

+ 1198623(119882lowastminus 1198821)119867

1198633119877119867

1

+ 1198624(119882lowastminus 1198821)119879

1198634119877119867

1]

= Re tr [11987711986711198601(119881lowastminus 1198811) 1198611+ 119877119867

11198602(119881lowast minus 119881

1)1198612

+ 119877119867

11198603(119881lowastminus 1198811)119867

1198613

+ 119877119867

11198604(119881lowastminus 1198811)119879

1198614

+ 119877119867

11198621(119882lowastminus 1198821)1198631

+ 119877119867

11198622(119882lowast minus 119882

1)1198632

+ 119877119867

11198623(119882lowastminus 1198821)119867

1198633

+ 119877119867

11198624(119882lowastminus 1198821)119879

1198634]

4 Journal of Discrete Mathematics

= Re tr [1198771198671(1198601119881lowast1198611+ 1198621119882lowast1198631+ 1198602119881lowast1198612

+ 1198622119882lowast1198632+ 1198603119881lowast119867

1198613+ 1198623119882lowast119867

1198633

+ 1198604119881lowast119879

1198614+ 1198624119882lowast119879

1198634minus 119860111988111198611

minus 119862111988211198631minus 119860211988111198612minus 119862211988211198632

minus 1198603119881119867

11198613minus 1198623119882119867

11198633

minus 1198604119881119879

11198614minus 1198624119882119879

11198634)]

(10)

In view that 119881lowast and 119882lowast are solutions of matrix equation (1)

with this relation we have

Re tr [1198751198671

(119881lowastminus 1198811) + 119876119867

1(119882lowastminus 1198821)]

= Re tr [1198771198671(119864 minus 119860

111988111198611minus 119862111988211198631

minus 119860211988111198612

minus 119862211988211198632

minus 1198603119881119867

11198613

minus 1198623119882119867

11198633

minus 1198604119881119879

11198614minus 1198624119882119879

11198634)]

= Re tr [11987711986711198771] =

100381710038171003817100381711987711003817100381710038171003817

2

(11)

This implies that (8) holds for 119894 = 1Assume that (8) holds for = 119896 That is

Re tr [119875119867119896

(119881lowastminus 119881119896) + 119876119867

119896(119882lowastminus 119882119896)] =

10038171003817100381710038171198771198961003817100381710038171003817

2

(12)

Then we have to prove that the conclusion holds for 119894 = 119896+ 1it follows from Algorithm 3 that

Re tr [119875119867119896+1

(119881lowastminus 119881119896+1

) + 119876119867

119896+1(119882lowastminus 119882119896+1

)]

= Re

tr[

[

(119860119867

1119877119896+1

119861119867

1+ 119860119867

2119877119896+1

119861119867

2

+ 1198613119877119867

119896+11198603+ 1198614119877119879

119896+11198604

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2119875119896)

119867

(119881lowastminus 119881119896+1

)

+ (119862119867

1119877119896+1

119863119867

1+ 119862119867

2119877119896+1

119863119867

2

+ 1198633119877119867

119896+11198623+ 1198634119877119879

119896+11198624

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2119876119896)

119867

(119882lowastminus 119882119896+1

)]

]

= Retr[119877119867119896+1

1198601(119881lowastminus 119881119896+1

) 1198611

+ 119877119879

119896+11198602(119881lowastminus 119881119896+1

) 1198612

+ 119877119896+1

119861119867

3(119881lowastminus 119881119896+1

) 119860119867

3

+ 119877119896+1

1198614

119867

(119881lowastminus 119881119896+1

) 1198604

119867

+ 119877119867

119896+11198621(119882lowastminus 119882119896+1

)1198631

+ 119877119879

119896+11198622(119882lowastminus 119882119896+1

)1198632

+ 119877119896+1

119863119867

3(119882lowastminus 119882119896+1

) 119862119867

3

+ 119877119896+1

1198634

119867

(119882lowastminus 119882119896+1

) 1198624

119867

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2(119875119867

119896(119881lowastminus 119881119896+1

)

+ 119876119867

119896(119882lowastminus 119882119896+1

))]

(13)

From properties of trace and conjugate we get

Re tr [119875119867119896+1

(119881lowastminus 119881119896+1

) + 119876119867

119896+1(119882lowastminus 119882119896+1

)]

= Retr[119877119867119896+1

1198601(119881lowastminus 119881119896+1

) 1198611

+ 119877119879

119896+11198602(119881lowast minus 119881

119896+1) 1198612

+ 1198603(119881lowastminus 119881119896+1

)119867

1198613119877119867

119896+1

+ 1198604(119881lowastminus 119881119896+1

)119879

1198614119877119867

119896+1

+ 119877119867

119896+11198621(119882lowastminus 119882119896+1

)1198631

+ 119877119879

119896+11198622(119882lowast minus 119882

119896+1)1198632

+ 1198623(119882lowastminus 119882119896+1

)119867

1198633119877119867

119896+1

+ 1198624(119882lowastminus 119882119896+1

)119879

1198634119877119867

119896+1+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2

times (119875119867

119896(119881lowastminus 119881119896+1

)

+ 119876119867

119896(119882lowastminus 119882119896+1

))]

Journal of Discrete Mathematics 5

= Re tr [119877119867119896+1

1198601(119881lowastminus 119881119896+1

) 1198611

+ 119877119867

119896+11198602(119881lowast minus 119881

119896+1)1198612

+ 119877119867

119896+11198603(119881lowastminus 119881119896+1

)119867

1198613

+ 119877119867

119896+11198604(119881lowastminus 119881119896+1

)119879

1198614

+ 119877119867

119896+11198621(119882lowastminus 119882119896+1

)1198631

+ 119877119867

119896+11198622(119882lowast minus 119882

119896+1)1198632

+ 119877119867

119896+11198623(119882lowastminus 119882119896+1

)119867

1198633

+ 119877119867

119896+11198624(119882lowastminus 119882119896+1

)119879

1198634] +

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2

times Retr[119875119867119896

(119881lowastminus 119881119896minus

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2119875119896)

+ 119876119867

119896(119882lowastminus 119882119896

minus

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2119876119896)]

= Re tr [119877119867119896+1

(1198601119881lowast1198611+ 1198621119882lowast1198631

+ 1198602119881lowast1198612+ 1198622119882lowast1198632

+ 1198603119881lowast119867

1198613+ 1198623119882lowast119867

1198633

+ 1198604119881lowast119879

1198614+ 1198624119882lowast119879

1198634

minus 1198601119881119896+1

1198611minus 1198621119882119896+1

1198631minus 1198602119881119896+1

1198612

minus 1198622119882119896+1

1198632

minus 1198603119881119867

119896+11198613

minus 1198623119882119867

119896+11198633minus 1198604119881119879

119896+11198614

minus 1198624119882119879

119896+11198634)]

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2Retr[119875119867

119896(119881lowastminus 119881119896) + 119876

119867

119896(119882lowastminus 119882119896)

minus

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2(119875119867

119896119875119896+ 119876119867

119896119876119896)]

(14)

In view that 119881lowast and 119882lowast are solutions of matrix equation (1)

with relation (14) one has

Re tr [119875119867119896+1

(119881lowastminus 119881119896+1

) + 119876119867

119896+1(119882lowastminus 119882119896+1

)]

= Re tr [119877119867119896+1

(119864 minus 1198601119881119896+1

1198611minus 1198621119882119896+1

1198631

minus 1198602119881119896+1

1198612minus 1198622119882119896+1

1198632

minus 1198603119881119867

119896+11198613minus 1198623119882119867

119896+11198633

minus 1198604119881119879

119896+11198614minus 1198624119882119879

119896+11198634)]

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2[1003817100381710038171003817119877119896

1003817100381710038171003817

2

minus

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2(1003817100381710038171003817119875119896

1003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

)]

= Re tr [119877119867119896+1

119877119896+1

] =1003817100381710038171003817119877119896+1

1003817100381710038171003817

2

(15)

Then relation (8) holds by mathematical induction

Lemma 5 Suppose that the matrix equation (1) is consis-tent and the sequences 119877

119894 119875119894 and 119876

119894 are generated

by Algorithm 3 with any initial matrices 1198811 1198821 such that

119877119894

= 0 for all 119894 = 1 2 119896 and then

Re tr (119877119867119895119877119894) = 0

Re tr (119875119867119895119875119894+ 119876119867

119895119876119894) = 0 119894 119895 = 1 2 119896 119894 = 119895

(16)

Proof We apply mathematical inductionStep 1 We prove that

Re tr (119877119867119894+1

119877119894) = 0 (17)

Re tr (119875119867119894+1

119875119894+ 119876119867

119894+1119876119894) = 0 119894 = 1 2 119896 (18)

First from Algorithm 3 we have

119877119896+1

= 119864 minus 1198601119881119896+1

1198611minus 1198621119882119896+1

1198631minus 1198602119881119896+1

1198612

minus 1198622119882119896+1

1198632minus 1198603119881119867

119896+11198613

minus 1198623119882119867

119896+11198633minus 1198604119881119879

119896+11198614minus 1198624119882119879

119896+11198634

119877119896+1

= 119864 minus 1198601(119881119896+

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2119875119896)1198611

minus 1198621(119882119896+

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2119876119896)1198631

minus 1198602(119881119896+

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2119875119896)1198612

minus 1198622(119882119896+

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2119876119896)1198632

minus 1198603(119881119896+

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2119875119896)

119867

1198613

minus 1198623(119882119896+

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2119876119896)

119867

1198633

minus 1198604(119881119896+

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2119875119896)

119879

1198614

6 Journal of Discrete Mathematics

minus 1198624(119882119896+

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2119876119896)

119879

1198634

= 119864 minus 11986011198811198961198611minus 11986211198821198961198631minus 11986021198811198961198612minus 11986221198821198961198632

minus 1198603119881119867

1198961198613minus 1198623119882119867

1198961198633minus 1198604119881119879

1198961198614

minus 1198624119882119879

1198961198634minus

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

times (11986011198751198961198611+ 11986211198761198961198631+ 11986021198751198961198612

+ 11986221198761198961198632+ 1198603119875119867

1198961198613

+ 1198623119876119867

1198961198633+ 1198604119875119879

1198961198614+ 1198624119876119879

1198961198634)

119877119896+1

= 119877119896minus

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

times (11986011198751198961198611+ 11986211198761198961198631+ 11986021198751198961198612

+ 11986221198761198961198632+ 1198603119875119867

1198961198613

+ 1198623119876119867

1198961198633+ 1198604119875119879

1198961198614+ 1198624119876119879

1198961198634)

(19)

For 119894 = 1 it follows from (19) that

Re tr (11987711986721198771)

= Re

tr[

[

(1198771minus

100381710038171003817100381711987711003817100381710038171003817

2

100381710038171003817100381711987511003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2

times (119860111987511198611+ 119862111987611198631

+ 119860211987511198612+ 119862211987611198632

+ 1198603119875119867

11198613+ 1198623119876119867

11198633

+ 1198604119875119879

11198614+ 1198624119876119879

11198634))

119867

1198771]

]

= Retr (11987711986711198771) minus

100381710038171003817100381711987711003817100381710038171003817

2

100381710038171003817100381711987511003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2

times tr (1198751198671119860119867

11198771119861119867

1+ 119876119867

1119862119867

11198771119863119867

1

+ 1198751

119867

119860119867

21198771119861119867

2+ 1198761

119867

119862119867

21198771119863119867

2

+ 1198751119860119867

31198771119861119867

3+ 1198761119862119867

31198771119863119867

3

+ 1198751119860119867

41198771119861119867

4+ 1198761119862119867

41198771119863119867

4)

=10038171003817100381710038171198771

1003817100381710038171003817

2

minus

100381710038171003817100381711987711003817100381710038171003817

2

100381710038171003817100381711987511003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2

times Re tr [1198751198671119860119867

11198771119861119867

1+ 119876119867

1119862119867

11198771119863119867

1

+ 1198751

119867

119860119867

21198771119861119867

2+ 1198761

119867

119862119867

21198771119863119867

2

+ 1198613119877119867

11198603119875119867

1+ 1198633119877119867

11198623119876119867

1

+ 1198614119877119879

11198604119875119867

1+ 1198634119877119879

11198624119876119867

1]

=10038171003817100381710038171198771

1003817100381710038171003817

2

minus

100381710038171003817100381711987711003817100381710038171003817

2

100381710038171003817100381711987511003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2

times Re tr [1198751198671

(119860119867

11198771119861119867

1+ 119860119867

21198771119861119867

2

+ 1198613119877119867

11198603+ 1198614119877119879

11198604)

+ 119876119867

1(119862119867

11198771119863119867

1+ 119862119867

21198771119863119867

2

+ 1198633119877119867

11198623+ 1198634119877119879

11198624)]

=10038171003817100381710038171198771

1003817100381710038171003817

2

minus

100381710038171003817100381711987711003817100381710038171003817

2

100381710038171003817100381711987511003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2Re tr (119875119867

11198751+ 119876119867

11198761)

Re tr (11987711986721198771)

=10038171003817100381710038171198771

1003817100381710038171003817

2

minus

100381710038171003817100381711987711003817100381710038171003817

2

100381710038171003817100381711987511003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2(10038171003817100381710038171198751

1003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2

) = 0

(20)

This implies that (17) is satisfied for 119894 = 1From Algorithm 3 we have

Re tr (11987511986721198751+ 119876119867

21198761)

= Re

tr[

[

(119860119867

11198772119861119867

1+ 119860119867

21198772119861119867

2+ 1198613119877119867

21198603

+ 1198614119877119879

21198604+

100381710038171003817100381711987721003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

21198751)

119867

1198751

+ (119862119867

11198772119863119867

1+ 119862119867

21198772119863119867

2+ 1198633119877119867

21198623

+1198634119877119879

21198624+

100381710038171003817100381711987721003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

21198761)

119867

1198761]

]

Journal of Discrete Mathematics 7

= Retr[1198771198672119860111987511198611+ 1198772

119867

119860211987511198612+ 1198772119861119867

31198751119860119867

3

+ 11987721198614

119867

11987511198604

119867

+

100381710038171003817100381711987721003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

2119875119867

11198751+ 119877119867

2119862111987611198631

+ 1198772

119867

119862211987611198632+ 1198772119863119867

31198761119862119867

3

+ 11987721198634

119867

11987611198624

119867

+

100381710038171003817100381711987721003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

2119876119867

11198761]

= Retr[1198771198672119860111987511198611+ 1198772

119867

119860211987511198612+ 1198603119875119867

11198613119877119867

2

+ 1198604119875119879

11198614119877119867

2+ 119877119867

2119862111987611198631

+ 1198772

119867

119862211987611198632+ 1198623119876119867

11198633119877119867

2

+ 1198624119876119879

11198634119877119867

2+

100381710038171003817100381711987721003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

2(119875119867

11198751+ 119876119867

11198761)]

= Retr[1198771198672119860111987511198611+ 119877119867

2119860211987511198612+ 119877119867

21198603119875119867

11198613

+ 119877119867

21198604119875119879

11198614+ 119877119867

2119862111987611198631

+ 119877119867

2119862211987611198632+ 119877119867

21198623119876119867

11198633

+ 119877119867

21198624119876119879

11198634+

100381710038171003817100381711987721003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

2(119875119867

11198751+ 119876119867

11198761)]

= Retr[1198771198672(119860111987511198611+ 119862111987611198631+ 119860211987511198612

+ 119862211987611198632+ 1198603119875119867

11198613+ 1198623119876119867

11198633

+1198604119875119879

11198614+ 1198624119876119867

11198634) ]

+

100381710038171003817100381711987721003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

2(10038171003817100381710038171198751

1003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2

)

=

100381710038171003817100381711987511003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

2Re tr [119877119867

2(1198771minus 1198772)]

+

100381710038171003817100381711987721003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

2(10038171003817100381710038171198751

1003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2

)

= minus

100381710038171003817100381711987511003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

2(10038171003817100381710038171198772

1003817100381710038171003817

2

)

+

100381710038171003817100381711987721003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

2(10038171003817100381710038171198751

1003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2

) = 0

(21)This implies that (18) is satisfied for 119894 = 1

Assume that (17) and (18) hold for 119894 = 119896 minus 1 fromAlgorithm 3 we have

Re tr (119877119867119896+1

119877119896)

= Re

tr[

[

(119877119896minus

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

times (11986011198751198961198611+ 11986211198761198961198631+ 11986021198751198961198612

+ 11986221198761198961198632+ 1198603119875119867

1198961198613+ 1198623119876119867

1198961198633

+ 1198604119875119879

1198961198614+ 1198624119876119879

1198961198634))

119867

119877119896]

]

Re tr (119877119867119896+1

119877119896)

= Retr (119877119867119896119877119896) minus

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

times tr (119875119867119896119860119867

1119877119896119861119867

1+ 119876119867

119896119862119867

1119877119896119863119867

1

+ 119875119896

119867

119860119867

2119877119896119861119867

2+ 119876119896

119867

119862119867

2119877119896119863119867

2

+ 119875119896119860119867

3119877119896119861119867

3+ 119876119896119862119867

3119877119896119863119867

3

+ 119875119896119860119867

4119877119896119861119867

4+ 119876119896119862119867

4119877119896119863119867

4)

=1003817100381710038171003817119877119896

1003817100381710038171003817

2

minus

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

times Re tr (119875119867119896119860119867

1119877119896119861119867

1+ 119876119867

119896119862119867

1119877119896119863119867

1

+ 119875119896

119867

119860119867

2119877119896119861119867

2+ 119876119896

119867

119862119867

2119877119896119863119867

2

+ 1198613119877119867

1198961198603119875119867

119896+ 1198633119877119867

1198961198623119876119867

119896

+ 1198614119877119879

1198961198604119875119867

119896+ 1198634119877119879

1198961198624119876119867

119896)

=10038171003817100381710038171198771

1003817100381710038171003817

2

minus

100381710038171003817100381711987711003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

times Re tr [119875119867119896

(119860119867

1119877119896119861119867

1+ 119860119867

2119877119896119861119867

2

+ 1198613119877119867

1198961198603+ 1198614119877119879

1198961198604)

+ 119876119867

119896(119862119867

1119877119896119863119867

1+ 119862119867

2119877119896119863119867

2

+ 1198633119877119867

1198961198623+ 1198634119877119879

1198961198624)]

8 Journal of Discrete Mathematics

=1003817100381710038171003817119877119896

1003817100381710038171003817

2

minus

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

times Retr[119875119867119896

(119875119896minus

10038171003817100381710038171198771198961003817100381710038171003817

2

1003817100381710038171003817119877119896minus11003817100381710038171003817

2119875119896minus1

)

+ 119876119867

119896(119876119896minus

10038171003817100381710038171198771198961003817100381710038171003817

2

1003817100381710038171003817119877119896minus11003817100381710038171003817

2119876119896minus1

)]

=1003817100381710038171003817119877119896

1003817100381710038171003817

2

minus

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2(1003817100381710038171003817119875119896

1003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

) = 0

(22)

Thus (17) holds for 119894 = 119896Also from Algorithm 3 we have

Re tr (119875119867119896+1

119875119896+ 119876119867

119896+1119876119896)

= Re

tr [

[

(119860119867

1119877119896+1

119861119867

1+ 119860119867

2119877119896+1

119861119867

2

+ 1198613119877119867

119896+11198603+ 1198614119877119879

119896+11198604

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2119875119896)

119867

119875119896

+ (119862119867

1119877119896+1

119863119867

1+ 119862119867

2119877119896+1

119863119867

2

+ 1198633119877119867

119896+11198623+ 1198634119877119879

119896+11198624

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2119876119896)

119867

119876119896]

]

= Retr[119877119867119896+1

11986011198751198961198611+ 119877119896+1

119867

11986021198751198961198612

+ 119877119896+1

119861119867

3119875119896119860119867

3+ 119877119896+1

1198614

119867

1198751198961198604

119867

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2119875119867

119896119875119896+ 119877119867

119896+111986211198761198961198631

+ 119877119896+1

119867

11986221198761198961198632+ 119877119896+1

119863119867

3119876119896119862119867

3

+ 119877119896+1

1198634

119867

1198761198961198624

119867

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2119876119867

119896119876119896]

= Retr[119877119867119896+1

11986011198751198961198611+ 119877119896+1

119867

11986021198751198961198612

+ 1198603119875119867

1198961198613119877119867

119896+1+ 1198604119875119879

1198961198614119877119867

119896+1

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2119875119867

119896119875119896+ 119877119867

119896+111986211198761198961198631

+ 119877119896+1

119867

11986221198761198961198632+ 1198623119876119867

1198961198633119877119867

119896+1

+ 1198624119876119879

1198961198634119877119867

119896+1+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2119876119867

119896119876119896]

= Retr[119877119867119896+1

(11986011198751198961198611+ 11986211198761198961198631

+ 11986021198751198961198612+ 11986221198761198961198632+ 1198603119875119867

1198961198613

+1198623119876119867

1198961198633+ 1198604119875119879

1198961198614+ 1198624119876119879

1198961198634)

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2(119875119867

119896119875119896+ 119876119867

119896119876119896)]

=

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2Re tr (119877119867

119896+1(119877119896minus 119877119896+1

))

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2(1003817100381710038171003817119875119896

1003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

)

= minus

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2(1003817100381710038171003817119877119896+1

1003817100381710038171003817

2

)

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2(1003817100381710038171003817119875119896

1003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

) = 0

(23)

This implies that (17) and (18) hold for 119894 = 119896Then relations (17) and (18) holds bymathematical induc-

tionStep 2 We want to show that

Re (tr (119877119867119894+119897119877119894)) = 0

Re (tr (119875119867119894+119897119875119894+ 119876119867

119894+119897119876119894)) = 0

(24)

holds for 119897 ge 1 We will prove this conclusion by inductionThe case of 119897 = 1 has been proven in Step 1 Now we assumethat (24) holds for 119897 le 119904 119904 ge 1 The aim is to show that

Re (tr (119877119867119894+119904+1

119877119894)) = 0

Re (tr (119875119867119894+119904+1

119875119894+ 119876119867

119894+119904+1119876119894)) = 0

(25)

First we prove the following

Re (tr (119877119867119904+1

1198770)) = 0

Re (tr (119875119867119904+1

1198750+ 119876119867

119904+11198760)) = 0

(26)

Journal of Discrete Mathematics 9

By using Algorithm 3 from (19) and induction we have

Re tr (119877119867119904+1

1198770)

= Re

tr[

[

(119877119904minus

10038171003817100381710038171198771199041003817100381710038171003817

2

10038171003817100381710038171198751199041003817100381710038171003817

2

+1003817100381710038171003817119876119904

1003817100381710038171003817

2

times (11986011198751199041198611+ 11986211198761199041198631+ 11986021198751199041198612

+ 11986221198761199041198632+ 1198603119875119867

1199041198613+ 1198623119876119867

1199041198633

+ 1198604119875119879

1199041198614+ 1198624119876119879

1199041198634))

119867

1198770]

]

= Re tr (1198771198671199041198770) minus

10038171003817100381710038171198771199041003817100381710038171003817

2

10038171003817100381710038171198751199041003817100381710038171003817

2

+1003817100381710038171003817119876119904

1003817100381710038171003817

2

times tr (119875119867119904119860119867

11198770119861119867

1+ 119876119867

119904119862119867

11198770119863119867

1

+ 119875119904

119867

119860119867

21198770119861119867

2+ 119876119904

119867

119862119867

21198770119863119867

2

+ 119875119904119860119867

31198770119861119867

3+ 119876119904119862119867

31198770119863119867

3

+ 119875119904119860119867

41198770119861119867

4+ 119876119904119862119867

41198770119863119867

4)

Re tr (119877119867119904+1

1198770)

= Re tr (1198771198671199041198770) minus

10038171003817100381710038171198771199041003817100381710038171003817

2

10038171003817100381710038171198751199041003817100381710038171003817

2

+1003817100381710038171003817119876119904

1003817100381710038171003817

2

times Re tr (119875119867119904119860119867

11198770119861119867

1+ 119876119867

119904119862119867

11198770119863119867

1

+ 119875119904

119867

119860119867

21198770119861119867

2+ 119876119904

119867

119862119867

21198770119863119867

2

+ 1198613119877119867

01198603119875119867

119904+ 1198633119877119867

01198623119876119867

119904

+ 1198614119877119879

01198604119875119867

119904+ 1198634119877119879

01198624119876119867

119904)

= minus

10038171003817100381710038171198771199041003817100381710038171003817

2

10038171003817100381710038171198751199041003817100381710038171003817

2

+1003817100381710038171003817119876119904

1003817100381710038171003817

2

times Re tr (119875119867119904

(119860119867

11198770119861119867

1+ 119860119867

21198770119861119867

2

+ 1198613119877119867

01198603+ 1198614119877119879

01198604)

+ 119876119867

119904(119862119867

11198770119863119867

1+ 119862119867

21198770119863119867

2

+1198633119877119867

01198623+ 1198634119877119879

01198624))

= minus

10038171003817100381710038171198771199041003817100381710038171003817

2

10038171003817100381710038171198751199041003817100381710038171003817

2

+1003817100381710038171003817119876119904

1003817100381710038171003817

2Re tr (119875119867

1199041198750+ 119876119867

1199041198760) = 0

Re tr (119875119867119904+1

1198750+ 119876119867

119904+11198760)

= Retr[(119860119867

1119877119904+1

119861119867

1+ 119860119867

2119877119904+1

119861119867

2

+ 1198613119877119867

119904+11198603+ 1198614119877119879

119904+11198604

+

1003817100381710038171003817119877119904+11003817100381710038171003817

2

10038171003817100381710038171198771199041003817100381710038171003817

2119875119904)

119867

1198750

+ (119862119867

1119877119904+1

119863119867

1+ 119862119867

2119877119904+1

119863119867

2

+ 1198633119877119867

119904+11198623+ 1198634119877119879

119896+11198624

+

1003817100381710038171003817119877119904+11003817100381710038171003817

2

10038171003817100381710038171198771199041003817100381710038171003817

2119876119904)

119867

1198760]

]

= Retr[119877119867119904+1

119860111987501198611+ 119877119904+1

119867

119860211987501198612

+ 119877119904+1

119861119867

31198750119860119867

3+ 119877119904+1

1198614

119867

11987501198604

119867

+

1003817100381710038171003817119877119904+11003817100381710038171003817

2

10038171003817100381710038171198771199041003817100381710038171003817

2119875119867

1199041198750+ 119877119867

119904+1119862111987601198631

+ 119877119904+1

119867

119862211987601198632+ 119877119904+1

119863119867

31198760119862119867

3

+ 119877119904+1

1198634

119867

11987601198624

119867

+

1003817100381710038171003817119877119904+11003817100381710038171003817

2

10038171003817100381710038171198771199041003817100381710038171003817

2119876119867

1199041198760]

= Retr[119877119867119904+1

119860111987501198611+ 119877119904+1

119867

119860211987501198612

+ 1198603119875119867

01198613119877119867

119904+1+ 1198604119875119879

01198614119877119867

119904+1

+

1003817100381710038171003817119877119904+11003817100381710038171003817

2

10038171003817100381710038171198771199041003817100381710038171003817

2119875119867

1199041198750+ 119877119867

119904+1119862111987601198631

+ 119877119904+1

119867

119862211987601198632+ 1198623119876119867

01198633119877119867

119904+1

+ 1198624119876119879

01198634119877119867

119904+1+

1003817100381710038171003817119877119904+11003817100381710038171003817

2

10038171003817100381710038171198771199041003817100381710038171003817

2119876119867

1199041198760]

= Retr[119877119867119904+1

(119860111987501198611+ 119862111987601198631+ 119860211987501198612

+ 119862211987601198632+ 1198603119875119867

01198613

+ 1198623119876119867

01198633+ 1198604119875119879

01198614

+ 1198624119876119879

01198634)

+

1003817100381710038171003817119877119904+11003817100381710038171003817

2

10038171003817100381710038171198771199041003817100381710038171003817

2(119875119867

1199041198750+ 119876119867

1199041198760)]

=

100381710038171003817100381711987501003817100381710038171003817

2

+10038171003817100381710038171198760

1003817100381710038171003817

2

100381710038171003817100381711987701003817100381710038171003817

2Re tr (119877119867

119904+1(1198770minus 1198771))= 0

(27)

Then (26) is holds

10 Journal of Discrete Mathematics

From Algorithm 3 we have

Re tr (119875119867119894+119904+1

119875119894+ 119876119867

119894+119904+1119876119894)

= Retr[(119860119867

1119877119894+119904+1

119861119867

1+ 119860119867

2119877119894+119904+1

119861119867

2

+ 1198613119877119867

119894+119904+11198603+ 1198614119877119879

119894+119904+11198604

+

1003817100381710038171003817119877119894+119904+11003817100381710038171003817

2

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2119875119894+119904

)

119867

119875119894

+ (119862119867

1119877119894+119904+1

119863119867

1+ 119862119867

2119877119894+119904+1

119863119867

2

+ 1198633119877119867

119894+119904+11198623+ 1198634119877119879

119894+119904+11198624

+

1003817100381710038171003817119877119894+119904+11003817100381710038171003817

2

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2119876119894+119904

)

119867

119876119894]

]

= Retr[119877119867119894+119904+1

11986011198751198941198611+ 119877119894+119904+1

119867

11986021198751198941198612

+ 119877119894+119904+1

119861119867

3119875119894119860119867

3+ 119877119894+119904+1

1198614

119867

1198751198941198604

119867

+

1003817100381710038171003817119877119894+119904+11003817100381710038171003817

2

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2119875119867

119894+119904119875119894+ 119877119867

119894+119904+111986211198761198941198631

+ 119877119894+119904+1

119867

11986221198761198941198632+ 119877119894+119904+1

119863119867

3119876119894119862119867

3

+ 119877119894+119904+1

1198634

119867

1198761198941198624

119867

+

1003817100381710038171003817119877119894+119904+11003817100381710038171003817

2

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2119876119867

119894+119904119876119894]

= Retr[119877119867119894+119904+1

11986011198751198941198611+ 119877119894+119904+1

119867

11986021198751198941198612

+ 1198603119875119867

1198941198613119877119867

119894+119904+1+ 1198604119875119879

1198941198614119877119867

119894+119904+1

+

1003817100381710038171003817119877119894+119904+11003817100381710038171003817

2

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2119875119867

119894+119904119875119894+ 119877119867

119894+119904+111986211198761198941198631

+ 119877119894+119904+1

119867

11986221198761198941198632+ 1198623119876119867

1198941198633119877119867

119894+119904+1

+ 1198624119876119879

1198961198634119877119867

119894+119904+1+

1003817100381710038171003817119877119894+119904+11003817100381710038171003817

2

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2119876119867

119894+119904119876119894]

= Retr[119877119867119894+119904+1

(11986011198751198941198611+ 11986211198761198941198631

+ 11986021198751198941198612+ 11986221198761198941198632

+ 1198603119875119867

1198941198613+ 1198623119875119867

1198941198633

+ 1198604119875119879

1198941198614+ 1198624119876119879

1198941198634)

+

1003817100381710038171003817119877119894+119904+11003817100381710038171003817

2

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2(119875119867

119894+119904119875119894+ 119876119867

119894+119904119876119894)]

=

10038171003817100381710038171198751198941003817100381710038171003817

2

+1003817100381710038171003817119876119894

1003817100381710038171003817

2

10038171003817100381710038171198771198941003817100381710038171003817

2Re tr (119877119867

119894+119904+1(119877119894minus 119877119894+1

))

=

10038171003817100381710038171198751198941003817100381710038171003817

2

+1003817100381710038171003817119876119894

1003817100381710038171003817

2

10038171003817100381710038171198771198941003817100381710038171003817

2Re tr (119877119867

119894+119904+1119877119894)

(28)

Also from (19) we have

Re tr (119877119867119894+119904+1

119877119894)

= Re

tr[

[

(119877119894+119904

minus

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2

1003817100381710038171003817119875119894+1199041003817100381710038171003817

2

+1003817100381710038171003817119876119894+119904

1003817100381710038171003817

2

times (1198601119875119894+119904

1198611+ 1198621119876119894+119904

1198631

+ 1198602119875119894+119904

1198612+ 1198622119876119894+119904

1198632

+ 1198603119875119867

119894+1199041198613+ 1198623119876119867

119894+1199041198633

+ 1198604119875119879

119894+1199041198614+ 1198624119876119879

119894+1199041198634))

119867

119877119894]

]

= Retr (119877119867119894+119904

119877119894) minus

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2

1003817100381710038171003817119875119894+1199041003817100381710038171003817

2

+1003817100381710038171003817119876119894+119904

1003817100381710038171003817

2

times tr (119875119867119894+119904

119860119867

1119877119894119861119867

1+ 119876119867

119894+119904119862119867

1119877119894119863119867

1

+ 119875119894+119904

119867

119860119867

2119877119894119861119867

2+ 119876119894+119904

119867

119862119867

2119877119894119863119867

2

+ 119875119894+119904

119860119867

3119877119894119861119867

3+ 119876119894+119904

119862119867

3119877119894119863119867

3

+ 119875119894+119904

119860119867

4119877119894119861119867

4+ 119876119894+119904

119862119867

4119877119894119863119867

4)

Re tr (119877119867119894+119904+1

119877119894)

= minus

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2

1003817100381710038171003817119875119894+1199041003817100381710038171003817

2

+1003817100381710038171003817119876119894+119904

1003817100381710038171003817

2

times Retr(119875119867119894+119904

119860119867

1119877119894119861119867

1+ 119876119867

119894+119904119862119867

1119877119894119863119867

1

+ 119875119894+119904

119867

119860119867

2119877119894119861119867

2+ 119876119894+119904

119867

119862119867

2119877119894119863119867

2

+ 1198613119877119867

1198941198603119875119867

119894+119904+ 1198633119877119867

1198941198623119876119867

119894+119904

+ 1198614119877119879

1198941198604119875119867

119894+119904+ 1198634119877119879

1198941198624119876119867

119894+119904)

Journal of Discrete Mathematics 11

= minus

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2

1003817100381710038171003817119875119894+1199041003817100381710038171003817

2

+1003817100381710038171003817119876119894+119904

1003817100381710038171003817

2

times Re tr [119875119867119894+119904

(119860119867

1119877119894119861119867

1+ 119860119867

2119877119894119861119867

2

+ 1198613119877119867

1198941198603+ 1198614119877119879

1198941198604)

+ 119876119867

119894+119904(119862119867

1119877119894119863119867

1+ 119862119867

2119877119894119863119867

2

+ 1198633119877119867

1198941198623+ 1198634119877119879

1198941198624)]

= minus

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2

1003817100381710038171003817119875119894+1199041003817100381710038171003817

2

+1003817100381710038171003817119876119894+119904

1003817100381710038171003817

2

times Re tr[119875119867119894+119904

(119875119894minus

10038171003817100381710038171198771198941003817100381710038171003817

2

1003817100381710038171003817119877119894minus11003817100381710038171003817

2119875119894minus1

)

+ 119876119867

119894+119904(119876119894minus

10038171003817100381710038171198771198941003817100381710038171003817

2

1003817100381710038171003817119877119894minus11003817100381710038171003817

2119876119894minus1

)]

=

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2

1003817100381710038171003817119875119894+1199041003817100381710038171003817

2

+1003817100381710038171003817119876119894+119904

1003817100381710038171003817

2

10038171003817100381710038171198771198941003817100381710038171003817

2

1003817100381710038171003817119877119894minus11003817100381710038171003817

2

times Re tr (119875119867119894+119904

119875119894minus1

+ 119876119867

119894+119904119876119894minus1

)

(29)

Repeating (28) and (29) one can easily obtain for certain 120572

and 120573

tr (119875119867119894+119904+1

119875119894+ 119876119867

119894+119904+1119876119894) = 120572 [tr (119875119867

119904+11198751+ 119876119867

119904+11198761)]

tr (119877119867119894+119904+1

119877119894) = 120573 [tr (119877119867

119904+11198771)]

(30)

Combining these two relations with (26) implies that (24)holds for 119897 = 119904 + 1 From Steps 1 and 2 the conclusion holdsby the principle of inductionWith the above two lemmas wehave the following theorem

Theorem 6 (see [32]) If the matrix equation (1) is consistentthen a solution can be obtained within finite iteration steps byusing Algorithm 3 for any initial matrices 119881

11198821

4 Numerical Example

In this section we present numerical example to illustrate theapplication of our proposed methods

Example 7 In this example we illustrate our theoreticalresults of Algorithm 3 for solving the system of matrixequation

11986011198811198611+ 11986211198821198631+ 11986021198811198612+ 11986221198821198632+ 11986031198811198671198613

+ 11986231198821198671198633+ 11986041198811198791198614+ 11986241198821198791198634= 119864

(31)

Because of the influence of the error of calculation theresidual 119877(119896) is usually unequal to zero in this process ofthe iteration We regard the matrix 119877(119896) as a zero matrix if119877(119896) lt 10

minus10Given

1198601= [

2 + 3119894 minus119894 1 + 119894

5 1 + 2119894 minus3]

1198602= [

2 + 3119894 minus119894 1 + 119894

5 1 + 2119894 minus3]

1198603= [

0 2 minus 119894 119894

minus1 + 3119894 2 0] 119860

4=[

0 1 minus 3119894 1 + 119894

0 4 + 119894 minus3119894]

1198621= [

1 + 2119894 3 minus 119894 4

minus119894 2119894 minus3] 119862

2=[

3 + 2119894 0 1 + 119894

0 4119894 1 minus 2119894]

1198623= [

1 minus 3119894 2119894 minus3119894

1 2 + 3119894 4119894] 119862

4=[

1 minus 2119894 0 2

3 minus 119894 1 + 119894 minus1]

1198611= [

[

4 + 119894 minus119894

0 1 minus 119894

4119894 2 + 2119894

]

]

1198612= [

[

0 119894

1 + 119894 0

minus1 minus 119894 3119894

]

]

1198613= [

[

0 1

minus3119894 4 + 119894

5 1 + 2119894

]

]

1198614=[

[

3 + 119894 minus1 minus 119894

0 2 minus 119894

minus1 + 119894 2

]

]

1198631= [

[

0 0

1 minus 3119894 minus119894

2119894 minus3119894

]

]

1198632= [

[

0 119894

1 + 119894 0

minus1 minus 119894 3119894

]

]

1198633= [

[

0 1

minus3119894 4 + 119894

5 1 + 2119894

]

]

1198634= [

[

3119894 minus2 + 119894

0 119894

minus2119894 minus4119894

]

]

119864 = [42 + 55119894 115 + 25119894

minus38 minus 119894 132 + 44119894]

(32)

Taking 1198811= [0 0

0 0

0 0

] and 1198821= [0 0

0 0] we apply Algorithm 3

to compute 119881119896119882119896

And iterating 42 steps we get

119881

=[

[

00126 + 18415119894 00827 + 06381119894 11221 minus 09428119894

minus06903 + 10185119894 18818 + 10203119894 09208 + 04569119894

05344 minus 04909119894 09280 + 07169119894 04872 minus 02734119894

]

]

119882

=[

[

04218 minus 09710119894 01763 + 05183119894 11331 minus 00432119894

06273 minus 01216119894 minus03902 + 04313119894 06240 + 09828119894

minus07011 minus 03418119894 03695 + 17627119894 minus03032 + 07073119894

]

]

(33)

which satisfy the matrix equation

11986011198811198611+ 11986211198821198631+ 11986021198811198612+ 11986221198821198632+ 11986031198811198671198613

+ 11986231198821198671198633+ 11986041198811198791198614+ 11986241198821198791198634= 119864

(34)

12 Journal of Discrete Mathematics

log10 rk

4

2

0

minus2

minus4

minus6

minus8

minus100 5 10 15 20 25 30 35 40 45

k (iteration number)

Figure 1The relation between the number of iterations and residualfor the example

With the corresponding residual100381710038171003817100381711987742

1003817100381710038171003817

=10038171003817100381710038171003817119864 minus 119860

1119881421198611minus 1198621119882421198631minus 1198602119881421198612minus 1198622119882421198632

minus 1198603119881119867

421198613minus 1198623119882119867

421198633minus 1198604119881119879

421198614minus 1198624119882119879

421198634

10038171003817100381710038171003817

= 66115 times 10minus11

(35)

5 Conclusions

The above Figure 1 shows the convergence curve for theresidual function 119877(119896) In this paper an iterative algorithmconstructed to solve a complex matrix equation with conju-gate and transpose of two unknowns of the form 119860

11198811198611+

11986211198821198631+11986021198811198612+11986221198821198632+11986031198811198671198613+11986231198821198671198633+11986041198811198791198614+

11986241198821198791198634

= 119864 is presented We proved that the iterativealgorithms always converge to the solution for any initialmatrices We stated and proved some lemmas and theoremswhere the solutions are obtained The proposed methodis illustrated by numerical example where the obtainednumerical results show that our technique is very neat andefficient

References

[1] F Ding P X Liu and J Ding ldquoIterative solutions of thegeneralized Sylvester matrix equations by using the hierarchicalidentification principlerdquo Applied Mathematics and Computa-tion vol 197 no 1 pp 41ndash50 2008

[2] F Ding and T Chen ldquoGradient based iterative algorithmsfor solving a class of matrix equationsrdquo IEEE Transactions onAutomatic Control vol 50 no 8 pp 1216ndash1221 2005

[3] FDing andTChen ldquoHierarchical gradient-based identificationof multivariable discrete-time systemsrdquo Automatica vol 41 no2 pp 315ndash325 2005

[4] F Ding and T Chen ldquoIterative least-squares solutions of cou-pled Sylvester matrix equationsrdquo Systems and Control Lettersvol 54 no 2 pp 95ndash107 2005

[5] F Ding and T Chen ldquoOn iterative solutions of general coupledmatrix equationsrdquo SIAM Journal on Control and Optimizationvol 44 no 6 pp 2269ndash2284 2006

[6] F Ding and T Chen ldquoHierarchical least squares identificationmethods for multivariable systemsrdquo IEEE Transactions onAutomatic Control vol 50 no 3 pp 397ndash402 2005

[7] F Ding and T Chen ldquoPerformance analysis ofmulti-innovationgradient type identification methodsrdquo Automatica vol 43 no1 pp 1ndash14 2007

[8] AGWuX ZengG RDuan andW JWu ldquoIterative solutionsto the extended Sylvester-conjugate matrix equationsrdquo AppliedMathematics and Computation vol 217 no 1 pp 130ndash142 2010

[9] B Zhou G R Duan and Z Y Li ldquoGradient based iterativealgorithm for solving coupled matrix equationsrdquo Systems andControl Letters vol 58 no 5 pp 327ndash333 2009

[10] J H Bevis F J Hall and R E Hartwing ldquoConsimilarity andthe matrix equation119860119883minus119883119861 = 119862rdquo in Current Trends inMatrixTheory pp 51ndash64 North-Holland New York NY USA 1987

[11] R A Horn and C R Johnson Matrix Analysis CambridgeUniversity Press Cambridge UK 1990

[12] H Liping ldquoConsimilarity of quaternion matrices and complexmatricesrdquo Linear Algebra and Its Applications vol 331 no 1ndash3pp 21ndash30 2001

[13] T Jiang X Cheng and L Chen ldquoAn algebraic relation betweenconsimilarity and similarity of complex matrices and its appli-cationsrdquo Journal of Physics A vol 39 no 29 pp 9215ndash92222006

[14] J H Bevis F J Hall and R E Hartwig ldquoThe matrix equation119860119883 minus 119883119861 = 119862 and its special casesrdquo SIAM Journal on MatrixAnalysis and Applications vol 9 no 3 pp 348ndash359 1988

[15] A G Wu G R Duan and H H Yu ldquoOn solutions of thematrix equations 119883119865 minus 119860119883 = 119862 and 119883119865 minus 119860119883 = 119862rdquo AppliedMathematics and Computation vol 183 no 2 pp 932ndash9412006

[16] A Navarra P L Odell and D M Young ldquoRepresentationof the general common solution to the matrix equations11986011198831198611= 1198621 11986021198831198612= 1198622with applicationsrdquo Computers and

Mathematics with Applications vol 41 no 7-8 pp 929ndash9352001

[17] J W Van der Woude ldquoOn the existence of a common solutionX to the matrix equations 119860

119894119883119861119895= 119862 (i j) E Irdquo Linear Algebra

and Its Applications vol 375 no 1ndash3 pp 135ndash145 2003[18] P Bhimasankaram ldquoCommon solutions to the linear matrix

equations 119860119883 = 119862119883119861 = 119863 and EXF = Grdquo Sankhya Series Avol 38 pp 404ndash409 1976

[19] S Kumar Mitra ldquoThe matrix equations 119860119883 = 119862119883119861 = 119863rdquoLinear Algebra and Its Applications vol 59 pp 171ndash181 1984

[20] S K Mitra ldquoA pair of simultaneous linear matrix equations11986011198831198611

= 1198621 11986021198831198612

= 1198622and a matrix programming

problemrdquo Linear Algebra and Its Applications vol 131 pp 107ndash123 1990

[21] M A Ramadan M A Abdel Naby and A M E BayoumildquoOn the explicit solutions of forms of the Sylvester and theYakubovich matrix equationsrdquo Mathematical and ComputerModelling vol 50 no 9-10 pp 1400ndash1408 2009

[22] K W E Chu ldquoSingular value and generalized singular valuedecompositions and the solution of linear matrix equationsrdquoLinear Algebra and Its Applications vol 88-89 pp 83ndash98 1987

Journal of Discrete Mathematics 13

[23] Y X Yuan ldquoThe optimal solution of linear matrix equationby matrix decompositions Mathrdquo Numerica Sinica vol 24 pp165ndash176 2002

[24] A P Liao and Y Lei ldquoLeast-squares solution with the mini-mum-norm for the matrix equation (119860119883119861 119866119883119867) = (119862119863)rdquoComputers and Mathematics with Applications vol 50 no 3-4pp 539ndash549 2005

[25] Y H Liu ldquoRanks of least squares solutions of the matrix equa-tion 119860119883119861 = 119862rdquo Computers and Mathematics with Applicationsvol 55 no 6 pp 1270ndash1278 2008

[26] X Sheng and G Chen ldquoA finite iterative method for solving apair of linear matrix equations (119860119883119861 119862119883119863) = (119864 119865)rdquo AppliedMathematics and Computation vol 189 no 2 pp 1350ndash13582007

[27] A G Wu W Liu and G R Duan ldquoOn the conjugate productof complex polynomial matricesrdquo Mathematical and ComputerModelling vol 53 no 9-10 pp 2031ndash2043 2011

[28] A GWu G FengW Liu and G R Duan ldquoThe complete solu-tion to the Sylvester-polynomial-conjugate matrix equationsrdquoMathematical and Computer Modelling vol 53 no 9-10 pp2044ndash2056 2011

[29] A G Wu B Li Y Zhang and G R Duan ldquoFinite iterativesolutions to coupled Sylvester-conjugate matrix equationsrdquoApplied Mathematical Modelling vol 35 no 3 pp 1065ndash10802011

[30] AGWu L Lv andGRDuan ldquoIterative algorithms for solvinga class of complex conjugate and transpose matrix equationsrdquoApplied Mathematics and Computation vol 217 no 21 pp8343ndash8353 2011

[31] X ZhangMatrix Analysis and Application TsinghuaUniversityPress Beijing China 2004

[32] A G Wu L Lv and M Z Hou ldquoFinite iterative algorithms forextended Sylvester-conjugate matrix equationsrdquo Mathematicaland Computer Modelling vol 54 pp 2363ndash2384 2011

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Differential EquationsInternational Journal of

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Stochastic AnalysisInternational Journal of

Journal of Discrete Mathematics 3

(3) Set

1198771= 119864 minus 119860

111988111198611minus 119862111988211198631minus 119860211988111198612

minus 119862211988211198632minus 1198603119881119867

11198613minus 1198623119882119867

11198633

minus 1198604119881119879

11198614minus 1198624119882119879

11198634

1198751= 119860119867

11198771119861119867

1+ 119860119867

21198771119861119867

2+ 1198613119877119867

11198603+ 1198614119877119879

11198604

1198761= 119862119867

11198771119863119867

1+ 119862119867

21198771119863119867

2+ 1198633119877119867

11198623+ 1198634119877119879

11198624

119896 = 1

(6)

(4) If 119877119896= 0 then stop else go to Step 5

(5) Set

119881119896+1

= 119881119896+

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2119875119896

119882119896+1

= 119882119896+

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2119876119896

119877119896+1

= 119864 minus 1198601119881119896+1

1198611minus 1198621119882119896+1

1198631minus 1198602119881119896+1

1198612

minus 1198622119882119896+1

1198632minus 1198603119881119867

119896+11198613minus 1198623119882119867

119896+11198633

minus 1198604119881119879

119896+11198614minus 1198624119882119879

119896+11198634

119875119896+1

= 119860119867

1119877119896+1

119861119867

1+ 119860119867

2119877119896+1

119861119867

2+ 1198613119877119867

119896+11198603

+ 1198614119877119879

119896+11198604+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2119875119896

119876119896+1

= 119862119867

1119877119896+1

119863119867

1+ 119862119867

2119877119896+1

119863119867

2

+ 1198633119877119867

119896+11198623+ 1198634119877119879

119896+11198624+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2119876119896

(7)

(6) If 119877119896+1

= 0 then stop else let 119896 = 119896 + 1 go to Step 5To prove the convergence property of Algorithm 3 we

first establish the following basic properties

Lemma 4 Suppose that the matrix equation (1) is consistentand 119881

lowast 119882lowast are arbitrary solutions of (1) Then for any initialmatrices 119881

1and119882

1 we have

Re tr [119875119867119894

(119881lowastminus 119881119894) + 119876119867

119894(119882lowastminus 119882119894)] =

10038171003817100381710038171198771198941003817100381710038171003817

2

(8)

where the sequence 119881119894 119875119894 119882119894 119876119894 and 119877

119894 are gener-

ated by Algorithm 3 for 119894 = 1 2

Proof We applymathematical induction to prove the conclu-sion

For 119894 = 1 from Algorithm 3 we have

Re tr [1198751198671

(119881lowastminus 1198811) + 119876119867

1(119882lowastminus 1198821)]

= Re tr [(11986011986711198771119861119867

1+ 119860119867

21198771119861119867

2

+ 1198613119877119867

11198603+ 1198614119877119879

11198604119861119867

2)

119867

(119881lowastminus 1198811)

+ (119862119867

11198771119863119867

1+ 119862119867

21198771119863119867

2

+1198633119877119867

11198623+ 1198634119877119879

11198624119863119867

2)

119867

times (119882lowastminus 1198821) ]

= Re tr [11987711986711198601(119881lowastminus 1198811) 1198611+ 1198771

119867

1198602(119881lowastminus 1198811) 1198612

+ 1198771119861119867

3(119881lowastminus 1198811) 119860119867

3+ 1198771119861119867

4(119881lowastminus 1198811) 119860119867

4

+ 119877119867

11198621(119882lowastminus 1198821)1198631+ 1198771

119867

1198622(119882lowastminus 1198821)1198632

+ 1198771119863119867

3(119882lowastminus 1198821) 119862119867

3

+ 1198771119863119867

4(119882lowastminus 1198821) 119862119867

4]

(9)

From properties of trace and conjugate

Re tr [1198751198671

(119881lowastminus 1198811) + 119876119867

1(119882lowastminus 1198821)]

= Retr [11987711986711198601(119881lowastminus 1198811) 1198611+ 1198771

119867

1198602(119881lowast minus 119881

1) 1198612

+ 1198603(119881lowastminus 1198811)119867

1198613119877119867

1+ 1198604(119881lowastminus 1198811)119879

1198614119877119867

1

+ 119877119867

11198621(119882lowastminus 1198821)1198631

+ 1198771

119867

1198622(119882lowast minus 119882

1)1198632

+ 1198623(119882lowastminus 1198821)119867

1198633119877119867

1

+ 1198624(119882lowastminus 1198821)119879

1198634119877119867

1]

= Re tr [11987711986711198601(119881lowastminus 1198811) 1198611+ 119877119867

11198602(119881lowast minus 119881

1)1198612

+ 119877119867

11198603(119881lowastminus 1198811)119867

1198613

+ 119877119867

11198604(119881lowastminus 1198811)119879

1198614

+ 119877119867

11198621(119882lowastminus 1198821)1198631

+ 119877119867

11198622(119882lowast minus 119882

1)1198632

+ 119877119867

11198623(119882lowastminus 1198821)119867

1198633

+ 119877119867

11198624(119882lowastminus 1198821)119879

1198634]

4 Journal of Discrete Mathematics

= Re tr [1198771198671(1198601119881lowast1198611+ 1198621119882lowast1198631+ 1198602119881lowast1198612

+ 1198622119882lowast1198632+ 1198603119881lowast119867

1198613+ 1198623119882lowast119867

1198633

+ 1198604119881lowast119879

1198614+ 1198624119882lowast119879

1198634minus 119860111988111198611

minus 119862111988211198631minus 119860211988111198612minus 119862211988211198632

minus 1198603119881119867

11198613minus 1198623119882119867

11198633

minus 1198604119881119879

11198614minus 1198624119882119879

11198634)]

(10)

In view that 119881lowast and 119882lowast are solutions of matrix equation (1)

with this relation we have

Re tr [1198751198671

(119881lowastminus 1198811) + 119876119867

1(119882lowastminus 1198821)]

= Re tr [1198771198671(119864 minus 119860

111988111198611minus 119862111988211198631

minus 119860211988111198612

minus 119862211988211198632

minus 1198603119881119867

11198613

minus 1198623119882119867

11198633

minus 1198604119881119879

11198614minus 1198624119882119879

11198634)]

= Re tr [11987711986711198771] =

100381710038171003817100381711987711003817100381710038171003817

2

(11)

This implies that (8) holds for 119894 = 1Assume that (8) holds for = 119896 That is

Re tr [119875119867119896

(119881lowastminus 119881119896) + 119876119867

119896(119882lowastminus 119882119896)] =

10038171003817100381710038171198771198961003817100381710038171003817

2

(12)

Then we have to prove that the conclusion holds for 119894 = 119896+ 1it follows from Algorithm 3 that

Re tr [119875119867119896+1

(119881lowastminus 119881119896+1

) + 119876119867

119896+1(119882lowastminus 119882119896+1

)]

= Re

tr[

[

(119860119867

1119877119896+1

119861119867

1+ 119860119867

2119877119896+1

119861119867

2

+ 1198613119877119867

119896+11198603+ 1198614119877119879

119896+11198604

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2119875119896)

119867

(119881lowastminus 119881119896+1

)

+ (119862119867

1119877119896+1

119863119867

1+ 119862119867

2119877119896+1

119863119867

2

+ 1198633119877119867

119896+11198623+ 1198634119877119879

119896+11198624

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2119876119896)

119867

(119882lowastminus 119882119896+1

)]

]

= Retr[119877119867119896+1

1198601(119881lowastminus 119881119896+1

) 1198611

+ 119877119879

119896+11198602(119881lowastminus 119881119896+1

) 1198612

+ 119877119896+1

119861119867

3(119881lowastminus 119881119896+1

) 119860119867

3

+ 119877119896+1

1198614

119867

(119881lowastminus 119881119896+1

) 1198604

119867

+ 119877119867

119896+11198621(119882lowastminus 119882119896+1

)1198631

+ 119877119879

119896+11198622(119882lowastminus 119882119896+1

)1198632

+ 119877119896+1

119863119867

3(119882lowastminus 119882119896+1

) 119862119867

3

+ 119877119896+1

1198634

119867

(119882lowastminus 119882119896+1

) 1198624

119867

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2(119875119867

119896(119881lowastminus 119881119896+1

)

+ 119876119867

119896(119882lowastminus 119882119896+1

))]

(13)

From properties of trace and conjugate we get

Re tr [119875119867119896+1

(119881lowastminus 119881119896+1

) + 119876119867

119896+1(119882lowastminus 119882119896+1

)]

= Retr[119877119867119896+1

1198601(119881lowastminus 119881119896+1

) 1198611

+ 119877119879

119896+11198602(119881lowast minus 119881

119896+1) 1198612

+ 1198603(119881lowastminus 119881119896+1

)119867

1198613119877119867

119896+1

+ 1198604(119881lowastminus 119881119896+1

)119879

1198614119877119867

119896+1

+ 119877119867

119896+11198621(119882lowastminus 119882119896+1

)1198631

+ 119877119879

119896+11198622(119882lowast minus 119882

119896+1)1198632

+ 1198623(119882lowastminus 119882119896+1

)119867

1198633119877119867

119896+1

+ 1198624(119882lowastminus 119882119896+1

)119879

1198634119877119867

119896+1+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2

times (119875119867

119896(119881lowastminus 119881119896+1

)

+ 119876119867

119896(119882lowastminus 119882119896+1

))]

Journal of Discrete Mathematics 5

= Re tr [119877119867119896+1

1198601(119881lowastminus 119881119896+1

) 1198611

+ 119877119867

119896+11198602(119881lowast minus 119881

119896+1)1198612

+ 119877119867

119896+11198603(119881lowastminus 119881119896+1

)119867

1198613

+ 119877119867

119896+11198604(119881lowastminus 119881119896+1

)119879

1198614

+ 119877119867

119896+11198621(119882lowastminus 119882119896+1

)1198631

+ 119877119867

119896+11198622(119882lowast minus 119882

119896+1)1198632

+ 119877119867

119896+11198623(119882lowastminus 119882119896+1

)119867

1198633

+ 119877119867

119896+11198624(119882lowastminus 119882119896+1

)119879

1198634] +

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2

times Retr[119875119867119896

(119881lowastminus 119881119896minus

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2119875119896)

+ 119876119867

119896(119882lowastminus 119882119896

minus

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2119876119896)]

= Re tr [119877119867119896+1

(1198601119881lowast1198611+ 1198621119882lowast1198631

+ 1198602119881lowast1198612+ 1198622119882lowast1198632

+ 1198603119881lowast119867

1198613+ 1198623119882lowast119867

1198633

+ 1198604119881lowast119879

1198614+ 1198624119882lowast119879

1198634

minus 1198601119881119896+1

1198611minus 1198621119882119896+1

1198631minus 1198602119881119896+1

1198612

minus 1198622119882119896+1

1198632

minus 1198603119881119867

119896+11198613

minus 1198623119882119867

119896+11198633minus 1198604119881119879

119896+11198614

minus 1198624119882119879

119896+11198634)]

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2Retr[119875119867

119896(119881lowastminus 119881119896) + 119876

119867

119896(119882lowastminus 119882119896)

minus

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2(119875119867

119896119875119896+ 119876119867

119896119876119896)]

(14)

In view that 119881lowast and 119882lowast are solutions of matrix equation (1)

with relation (14) one has

Re tr [119875119867119896+1

(119881lowastminus 119881119896+1

) + 119876119867

119896+1(119882lowastminus 119882119896+1

)]

= Re tr [119877119867119896+1

(119864 minus 1198601119881119896+1

1198611minus 1198621119882119896+1

1198631

minus 1198602119881119896+1

1198612minus 1198622119882119896+1

1198632

minus 1198603119881119867

119896+11198613minus 1198623119882119867

119896+11198633

minus 1198604119881119879

119896+11198614minus 1198624119882119879

119896+11198634)]

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2[1003817100381710038171003817119877119896

1003817100381710038171003817

2

minus

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2(1003817100381710038171003817119875119896

1003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

)]

= Re tr [119877119867119896+1

119877119896+1

] =1003817100381710038171003817119877119896+1

1003817100381710038171003817

2

(15)

Then relation (8) holds by mathematical induction

Lemma 5 Suppose that the matrix equation (1) is consis-tent and the sequences 119877

119894 119875119894 and 119876

119894 are generated

by Algorithm 3 with any initial matrices 1198811 1198821 such that

119877119894

= 0 for all 119894 = 1 2 119896 and then

Re tr (119877119867119895119877119894) = 0

Re tr (119875119867119895119875119894+ 119876119867

119895119876119894) = 0 119894 119895 = 1 2 119896 119894 = 119895

(16)

Proof We apply mathematical inductionStep 1 We prove that

Re tr (119877119867119894+1

119877119894) = 0 (17)

Re tr (119875119867119894+1

119875119894+ 119876119867

119894+1119876119894) = 0 119894 = 1 2 119896 (18)

First from Algorithm 3 we have

119877119896+1

= 119864 minus 1198601119881119896+1

1198611minus 1198621119882119896+1

1198631minus 1198602119881119896+1

1198612

minus 1198622119882119896+1

1198632minus 1198603119881119867

119896+11198613

minus 1198623119882119867

119896+11198633minus 1198604119881119879

119896+11198614minus 1198624119882119879

119896+11198634

119877119896+1

= 119864 minus 1198601(119881119896+

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2119875119896)1198611

minus 1198621(119882119896+

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2119876119896)1198631

minus 1198602(119881119896+

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2119875119896)1198612

minus 1198622(119882119896+

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2119876119896)1198632

minus 1198603(119881119896+

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2119875119896)

119867

1198613

minus 1198623(119882119896+

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2119876119896)

119867

1198633

minus 1198604(119881119896+

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2119875119896)

119879

1198614

6 Journal of Discrete Mathematics

minus 1198624(119882119896+

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2119876119896)

119879

1198634

= 119864 minus 11986011198811198961198611minus 11986211198821198961198631minus 11986021198811198961198612minus 11986221198821198961198632

minus 1198603119881119867

1198961198613minus 1198623119882119867

1198961198633minus 1198604119881119879

1198961198614

minus 1198624119882119879

1198961198634minus

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

times (11986011198751198961198611+ 11986211198761198961198631+ 11986021198751198961198612

+ 11986221198761198961198632+ 1198603119875119867

1198961198613

+ 1198623119876119867

1198961198633+ 1198604119875119879

1198961198614+ 1198624119876119879

1198961198634)

119877119896+1

= 119877119896minus

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

times (11986011198751198961198611+ 11986211198761198961198631+ 11986021198751198961198612

+ 11986221198761198961198632+ 1198603119875119867

1198961198613

+ 1198623119876119867

1198961198633+ 1198604119875119879

1198961198614+ 1198624119876119879

1198961198634)

(19)

For 119894 = 1 it follows from (19) that

Re tr (11987711986721198771)

= Re

tr[

[

(1198771minus

100381710038171003817100381711987711003817100381710038171003817

2

100381710038171003817100381711987511003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2

times (119860111987511198611+ 119862111987611198631

+ 119860211987511198612+ 119862211987611198632

+ 1198603119875119867

11198613+ 1198623119876119867

11198633

+ 1198604119875119879

11198614+ 1198624119876119879

11198634))

119867

1198771]

]

= Retr (11987711986711198771) minus

100381710038171003817100381711987711003817100381710038171003817

2

100381710038171003817100381711987511003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2

times tr (1198751198671119860119867

11198771119861119867

1+ 119876119867

1119862119867

11198771119863119867

1

+ 1198751

119867

119860119867

21198771119861119867

2+ 1198761

119867

119862119867

21198771119863119867

2

+ 1198751119860119867

31198771119861119867

3+ 1198761119862119867

31198771119863119867

3

+ 1198751119860119867

41198771119861119867

4+ 1198761119862119867

41198771119863119867

4)

=10038171003817100381710038171198771

1003817100381710038171003817

2

minus

100381710038171003817100381711987711003817100381710038171003817

2

100381710038171003817100381711987511003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2

times Re tr [1198751198671119860119867

11198771119861119867

1+ 119876119867

1119862119867

11198771119863119867

1

+ 1198751

119867

119860119867

21198771119861119867

2+ 1198761

119867

119862119867

21198771119863119867

2

+ 1198613119877119867

11198603119875119867

1+ 1198633119877119867

11198623119876119867

1

+ 1198614119877119879

11198604119875119867

1+ 1198634119877119879

11198624119876119867

1]

=10038171003817100381710038171198771

1003817100381710038171003817

2

minus

100381710038171003817100381711987711003817100381710038171003817

2

100381710038171003817100381711987511003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2

times Re tr [1198751198671

(119860119867

11198771119861119867

1+ 119860119867

21198771119861119867

2

+ 1198613119877119867

11198603+ 1198614119877119879

11198604)

+ 119876119867

1(119862119867

11198771119863119867

1+ 119862119867

21198771119863119867

2

+ 1198633119877119867

11198623+ 1198634119877119879

11198624)]

=10038171003817100381710038171198771

1003817100381710038171003817

2

minus

100381710038171003817100381711987711003817100381710038171003817

2

100381710038171003817100381711987511003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2Re tr (119875119867

11198751+ 119876119867

11198761)

Re tr (11987711986721198771)

=10038171003817100381710038171198771

1003817100381710038171003817

2

minus

100381710038171003817100381711987711003817100381710038171003817

2

100381710038171003817100381711987511003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2(10038171003817100381710038171198751

1003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2

) = 0

(20)

This implies that (17) is satisfied for 119894 = 1From Algorithm 3 we have

Re tr (11987511986721198751+ 119876119867

21198761)

= Re

tr[

[

(119860119867

11198772119861119867

1+ 119860119867

21198772119861119867

2+ 1198613119877119867

21198603

+ 1198614119877119879

21198604+

100381710038171003817100381711987721003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

21198751)

119867

1198751

+ (119862119867

11198772119863119867

1+ 119862119867

21198772119863119867

2+ 1198633119877119867

21198623

+1198634119877119879

21198624+

100381710038171003817100381711987721003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

21198761)

119867

1198761]

]

Journal of Discrete Mathematics 7

= Retr[1198771198672119860111987511198611+ 1198772

119867

119860211987511198612+ 1198772119861119867

31198751119860119867

3

+ 11987721198614

119867

11987511198604

119867

+

100381710038171003817100381711987721003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

2119875119867

11198751+ 119877119867

2119862111987611198631

+ 1198772

119867

119862211987611198632+ 1198772119863119867

31198761119862119867

3

+ 11987721198634

119867

11987611198624

119867

+

100381710038171003817100381711987721003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

2119876119867

11198761]

= Retr[1198771198672119860111987511198611+ 1198772

119867

119860211987511198612+ 1198603119875119867

11198613119877119867

2

+ 1198604119875119879

11198614119877119867

2+ 119877119867

2119862111987611198631

+ 1198772

119867

119862211987611198632+ 1198623119876119867

11198633119877119867

2

+ 1198624119876119879

11198634119877119867

2+

100381710038171003817100381711987721003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

2(119875119867

11198751+ 119876119867

11198761)]

= Retr[1198771198672119860111987511198611+ 119877119867

2119860211987511198612+ 119877119867

21198603119875119867

11198613

+ 119877119867

21198604119875119879

11198614+ 119877119867

2119862111987611198631

+ 119877119867

2119862211987611198632+ 119877119867

21198623119876119867

11198633

+ 119877119867

21198624119876119879

11198634+

100381710038171003817100381711987721003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

2(119875119867

11198751+ 119876119867

11198761)]

= Retr[1198771198672(119860111987511198611+ 119862111987611198631+ 119860211987511198612

+ 119862211987611198632+ 1198603119875119867

11198613+ 1198623119876119867

11198633

+1198604119875119879

11198614+ 1198624119876119867

11198634) ]

+

100381710038171003817100381711987721003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

2(10038171003817100381710038171198751

1003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2

)

=

100381710038171003817100381711987511003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

2Re tr [119877119867

2(1198771minus 1198772)]

+

100381710038171003817100381711987721003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

2(10038171003817100381710038171198751

1003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2

)

= minus

100381710038171003817100381711987511003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

2(10038171003817100381710038171198772

1003817100381710038171003817

2

)

+

100381710038171003817100381711987721003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

2(10038171003817100381710038171198751

1003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2

) = 0

(21)This implies that (18) is satisfied for 119894 = 1

Assume that (17) and (18) hold for 119894 = 119896 minus 1 fromAlgorithm 3 we have

Re tr (119877119867119896+1

119877119896)

= Re

tr[

[

(119877119896minus

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

times (11986011198751198961198611+ 11986211198761198961198631+ 11986021198751198961198612

+ 11986221198761198961198632+ 1198603119875119867

1198961198613+ 1198623119876119867

1198961198633

+ 1198604119875119879

1198961198614+ 1198624119876119879

1198961198634))

119867

119877119896]

]

Re tr (119877119867119896+1

119877119896)

= Retr (119877119867119896119877119896) minus

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

times tr (119875119867119896119860119867

1119877119896119861119867

1+ 119876119867

119896119862119867

1119877119896119863119867

1

+ 119875119896

119867

119860119867

2119877119896119861119867

2+ 119876119896

119867

119862119867

2119877119896119863119867

2

+ 119875119896119860119867

3119877119896119861119867

3+ 119876119896119862119867

3119877119896119863119867

3

+ 119875119896119860119867

4119877119896119861119867

4+ 119876119896119862119867

4119877119896119863119867

4)

=1003817100381710038171003817119877119896

1003817100381710038171003817

2

minus

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

times Re tr (119875119867119896119860119867

1119877119896119861119867

1+ 119876119867

119896119862119867

1119877119896119863119867

1

+ 119875119896

119867

119860119867

2119877119896119861119867

2+ 119876119896

119867

119862119867

2119877119896119863119867

2

+ 1198613119877119867

1198961198603119875119867

119896+ 1198633119877119867

1198961198623119876119867

119896

+ 1198614119877119879

1198961198604119875119867

119896+ 1198634119877119879

1198961198624119876119867

119896)

=10038171003817100381710038171198771

1003817100381710038171003817

2

minus

100381710038171003817100381711987711003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

times Re tr [119875119867119896

(119860119867

1119877119896119861119867

1+ 119860119867

2119877119896119861119867

2

+ 1198613119877119867

1198961198603+ 1198614119877119879

1198961198604)

+ 119876119867

119896(119862119867

1119877119896119863119867

1+ 119862119867

2119877119896119863119867

2

+ 1198633119877119867

1198961198623+ 1198634119877119879

1198961198624)]

8 Journal of Discrete Mathematics

=1003817100381710038171003817119877119896

1003817100381710038171003817

2

minus

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

times Retr[119875119867119896

(119875119896minus

10038171003817100381710038171198771198961003817100381710038171003817

2

1003817100381710038171003817119877119896minus11003817100381710038171003817

2119875119896minus1

)

+ 119876119867

119896(119876119896minus

10038171003817100381710038171198771198961003817100381710038171003817

2

1003817100381710038171003817119877119896minus11003817100381710038171003817

2119876119896minus1

)]

=1003817100381710038171003817119877119896

1003817100381710038171003817

2

minus

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2(1003817100381710038171003817119875119896

1003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

) = 0

(22)

Thus (17) holds for 119894 = 119896Also from Algorithm 3 we have

Re tr (119875119867119896+1

119875119896+ 119876119867

119896+1119876119896)

= Re

tr [

[

(119860119867

1119877119896+1

119861119867

1+ 119860119867

2119877119896+1

119861119867

2

+ 1198613119877119867

119896+11198603+ 1198614119877119879

119896+11198604

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2119875119896)

119867

119875119896

+ (119862119867

1119877119896+1

119863119867

1+ 119862119867

2119877119896+1

119863119867

2

+ 1198633119877119867

119896+11198623+ 1198634119877119879

119896+11198624

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2119876119896)

119867

119876119896]

]

= Retr[119877119867119896+1

11986011198751198961198611+ 119877119896+1

119867

11986021198751198961198612

+ 119877119896+1

119861119867

3119875119896119860119867

3+ 119877119896+1

1198614

119867

1198751198961198604

119867

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2119875119867

119896119875119896+ 119877119867

119896+111986211198761198961198631

+ 119877119896+1

119867

11986221198761198961198632+ 119877119896+1

119863119867

3119876119896119862119867

3

+ 119877119896+1

1198634

119867

1198761198961198624

119867

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2119876119867

119896119876119896]

= Retr[119877119867119896+1

11986011198751198961198611+ 119877119896+1

119867

11986021198751198961198612

+ 1198603119875119867

1198961198613119877119867

119896+1+ 1198604119875119879

1198961198614119877119867

119896+1

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2119875119867

119896119875119896+ 119877119867

119896+111986211198761198961198631

+ 119877119896+1

119867

11986221198761198961198632+ 1198623119876119867

1198961198633119877119867

119896+1

+ 1198624119876119879

1198961198634119877119867

119896+1+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2119876119867

119896119876119896]

= Retr[119877119867119896+1

(11986011198751198961198611+ 11986211198761198961198631

+ 11986021198751198961198612+ 11986221198761198961198632+ 1198603119875119867

1198961198613

+1198623119876119867

1198961198633+ 1198604119875119879

1198961198614+ 1198624119876119879

1198961198634)

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2(119875119867

119896119875119896+ 119876119867

119896119876119896)]

=

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2Re tr (119877119867

119896+1(119877119896minus 119877119896+1

))

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2(1003817100381710038171003817119875119896

1003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

)

= minus

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2(1003817100381710038171003817119877119896+1

1003817100381710038171003817

2

)

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2(1003817100381710038171003817119875119896

1003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

) = 0

(23)

This implies that (17) and (18) hold for 119894 = 119896Then relations (17) and (18) holds bymathematical induc-

tionStep 2 We want to show that

Re (tr (119877119867119894+119897119877119894)) = 0

Re (tr (119875119867119894+119897119875119894+ 119876119867

119894+119897119876119894)) = 0

(24)

holds for 119897 ge 1 We will prove this conclusion by inductionThe case of 119897 = 1 has been proven in Step 1 Now we assumethat (24) holds for 119897 le 119904 119904 ge 1 The aim is to show that

Re (tr (119877119867119894+119904+1

119877119894)) = 0

Re (tr (119875119867119894+119904+1

119875119894+ 119876119867

119894+119904+1119876119894)) = 0

(25)

First we prove the following

Re (tr (119877119867119904+1

1198770)) = 0

Re (tr (119875119867119904+1

1198750+ 119876119867

119904+11198760)) = 0

(26)

Journal of Discrete Mathematics 9

By using Algorithm 3 from (19) and induction we have

Re tr (119877119867119904+1

1198770)

= Re

tr[

[

(119877119904minus

10038171003817100381710038171198771199041003817100381710038171003817

2

10038171003817100381710038171198751199041003817100381710038171003817

2

+1003817100381710038171003817119876119904

1003817100381710038171003817

2

times (11986011198751199041198611+ 11986211198761199041198631+ 11986021198751199041198612

+ 11986221198761199041198632+ 1198603119875119867

1199041198613+ 1198623119876119867

1199041198633

+ 1198604119875119879

1199041198614+ 1198624119876119879

1199041198634))

119867

1198770]

]

= Re tr (1198771198671199041198770) minus

10038171003817100381710038171198771199041003817100381710038171003817

2

10038171003817100381710038171198751199041003817100381710038171003817

2

+1003817100381710038171003817119876119904

1003817100381710038171003817

2

times tr (119875119867119904119860119867

11198770119861119867

1+ 119876119867

119904119862119867

11198770119863119867

1

+ 119875119904

119867

119860119867

21198770119861119867

2+ 119876119904

119867

119862119867

21198770119863119867

2

+ 119875119904119860119867

31198770119861119867

3+ 119876119904119862119867

31198770119863119867

3

+ 119875119904119860119867

41198770119861119867

4+ 119876119904119862119867

41198770119863119867

4)

Re tr (119877119867119904+1

1198770)

= Re tr (1198771198671199041198770) minus

10038171003817100381710038171198771199041003817100381710038171003817

2

10038171003817100381710038171198751199041003817100381710038171003817

2

+1003817100381710038171003817119876119904

1003817100381710038171003817

2

times Re tr (119875119867119904119860119867

11198770119861119867

1+ 119876119867

119904119862119867

11198770119863119867

1

+ 119875119904

119867

119860119867

21198770119861119867

2+ 119876119904

119867

119862119867

21198770119863119867

2

+ 1198613119877119867

01198603119875119867

119904+ 1198633119877119867

01198623119876119867

119904

+ 1198614119877119879

01198604119875119867

119904+ 1198634119877119879

01198624119876119867

119904)

= minus

10038171003817100381710038171198771199041003817100381710038171003817

2

10038171003817100381710038171198751199041003817100381710038171003817

2

+1003817100381710038171003817119876119904

1003817100381710038171003817

2

times Re tr (119875119867119904

(119860119867

11198770119861119867

1+ 119860119867

21198770119861119867

2

+ 1198613119877119867

01198603+ 1198614119877119879

01198604)

+ 119876119867

119904(119862119867

11198770119863119867

1+ 119862119867

21198770119863119867

2

+1198633119877119867

01198623+ 1198634119877119879

01198624))

= minus

10038171003817100381710038171198771199041003817100381710038171003817

2

10038171003817100381710038171198751199041003817100381710038171003817

2

+1003817100381710038171003817119876119904

1003817100381710038171003817

2Re tr (119875119867

1199041198750+ 119876119867

1199041198760) = 0

Re tr (119875119867119904+1

1198750+ 119876119867

119904+11198760)

= Retr[(119860119867

1119877119904+1

119861119867

1+ 119860119867

2119877119904+1

119861119867

2

+ 1198613119877119867

119904+11198603+ 1198614119877119879

119904+11198604

+

1003817100381710038171003817119877119904+11003817100381710038171003817

2

10038171003817100381710038171198771199041003817100381710038171003817

2119875119904)

119867

1198750

+ (119862119867

1119877119904+1

119863119867

1+ 119862119867

2119877119904+1

119863119867

2

+ 1198633119877119867

119904+11198623+ 1198634119877119879

119896+11198624

+

1003817100381710038171003817119877119904+11003817100381710038171003817

2

10038171003817100381710038171198771199041003817100381710038171003817

2119876119904)

119867

1198760]

]

= Retr[119877119867119904+1

119860111987501198611+ 119877119904+1

119867

119860211987501198612

+ 119877119904+1

119861119867

31198750119860119867

3+ 119877119904+1

1198614

119867

11987501198604

119867

+

1003817100381710038171003817119877119904+11003817100381710038171003817

2

10038171003817100381710038171198771199041003817100381710038171003817

2119875119867

1199041198750+ 119877119867

119904+1119862111987601198631

+ 119877119904+1

119867

119862211987601198632+ 119877119904+1

119863119867

31198760119862119867

3

+ 119877119904+1

1198634

119867

11987601198624

119867

+

1003817100381710038171003817119877119904+11003817100381710038171003817

2

10038171003817100381710038171198771199041003817100381710038171003817

2119876119867

1199041198760]

= Retr[119877119867119904+1

119860111987501198611+ 119877119904+1

119867

119860211987501198612

+ 1198603119875119867

01198613119877119867

119904+1+ 1198604119875119879

01198614119877119867

119904+1

+

1003817100381710038171003817119877119904+11003817100381710038171003817

2

10038171003817100381710038171198771199041003817100381710038171003817

2119875119867

1199041198750+ 119877119867

119904+1119862111987601198631

+ 119877119904+1

119867

119862211987601198632+ 1198623119876119867

01198633119877119867

119904+1

+ 1198624119876119879

01198634119877119867

119904+1+

1003817100381710038171003817119877119904+11003817100381710038171003817

2

10038171003817100381710038171198771199041003817100381710038171003817

2119876119867

1199041198760]

= Retr[119877119867119904+1

(119860111987501198611+ 119862111987601198631+ 119860211987501198612

+ 119862211987601198632+ 1198603119875119867

01198613

+ 1198623119876119867

01198633+ 1198604119875119879

01198614

+ 1198624119876119879

01198634)

+

1003817100381710038171003817119877119904+11003817100381710038171003817

2

10038171003817100381710038171198771199041003817100381710038171003817

2(119875119867

1199041198750+ 119876119867

1199041198760)]

=

100381710038171003817100381711987501003817100381710038171003817

2

+10038171003817100381710038171198760

1003817100381710038171003817

2

100381710038171003817100381711987701003817100381710038171003817

2Re tr (119877119867

119904+1(1198770minus 1198771))= 0

(27)

Then (26) is holds

10 Journal of Discrete Mathematics

From Algorithm 3 we have

Re tr (119875119867119894+119904+1

119875119894+ 119876119867

119894+119904+1119876119894)

= Retr[(119860119867

1119877119894+119904+1

119861119867

1+ 119860119867

2119877119894+119904+1

119861119867

2

+ 1198613119877119867

119894+119904+11198603+ 1198614119877119879

119894+119904+11198604

+

1003817100381710038171003817119877119894+119904+11003817100381710038171003817

2

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2119875119894+119904

)

119867

119875119894

+ (119862119867

1119877119894+119904+1

119863119867

1+ 119862119867

2119877119894+119904+1

119863119867

2

+ 1198633119877119867

119894+119904+11198623+ 1198634119877119879

119894+119904+11198624

+

1003817100381710038171003817119877119894+119904+11003817100381710038171003817

2

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2119876119894+119904

)

119867

119876119894]

]

= Retr[119877119867119894+119904+1

11986011198751198941198611+ 119877119894+119904+1

119867

11986021198751198941198612

+ 119877119894+119904+1

119861119867

3119875119894119860119867

3+ 119877119894+119904+1

1198614

119867

1198751198941198604

119867

+

1003817100381710038171003817119877119894+119904+11003817100381710038171003817

2

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2119875119867

119894+119904119875119894+ 119877119867

119894+119904+111986211198761198941198631

+ 119877119894+119904+1

119867

11986221198761198941198632+ 119877119894+119904+1

119863119867

3119876119894119862119867

3

+ 119877119894+119904+1

1198634

119867

1198761198941198624

119867

+

1003817100381710038171003817119877119894+119904+11003817100381710038171003817

2

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2119876119867

119894+119904119876119894]

= Retr[119877119867119894+119904+1

11986011198751198941198611+ 119877119894+119904+1

119867

11986021198751198941198612

+ 1198603119875119867

1198941198613119877119867

119894+119904+1+ 1198604119875119879

1198941198614119877119867

119894+119904+1

+

1003817100381710038171003817119877119894+119904+11003817100381710038171003817

2

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2119875119867

119894+119904119875119894+ 119877119867

119894+119904+111986211198761198941198631

+ 119877119894+119904+1

119867

11986221198761198941198632+ 1198623119876119867

1198941198633119877119867

119894+119904+1

+ 1198624119876119879

1198961198634119877119867

119894+119904+1+

1003817100381710038171003817119877119894+119904+11003817100381710038171003817

2

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2119876119867

119894+119904119876119894]

= Retr[119877119867119894+119904+1

(11986011198751198941198611+ 11986211198761198941198631

+ 11986021198751198941198612+ 11986221198761198941198632

+ 1198603119875119867

1198941198613+ 1198623119875119867

1198941198633

+ 1198604119875119879

1198941198614+ 1198624119876119879

1198941198634)

+

1003817100381710038171003817119877119894+119904+11003817100381710038171003817

2

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2(119875119867

119894+119904119875119894+ 119876119867

119894+119904119876119894)]

=

10038171003817100381710038171198751198941003817100381710038171003817

2

+1003817100381710038171003817119876119894

1003817100381710038171003817

2

10038171003817100381710038171198771198941003817100381710038171003817

2Re tr (119877119867

119894+119904+1(119877119894minus 119877119894+1

))

=

10038171003817100381710038171198751198941003817100381710038171003817

2

+1003817100381710038171003817119876119894

1003817100381710038171003817

2

10038171003817100381710038171198771198941003817100381710038171003817

2Re tr (119877119867

119894+119904+1119877119894)

(28)

Also from (19) we have

Re tr (119877119867119894+119904+1

119877119894)

= Re

tr[

[

(119877119894+119904

minus

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2

1003817100381710038171003817119875119894+1199041003817100381710038171003817

2

+1003817100381710038171003817119876119894+119904

1003817100381710038171003817

2

times (1198601119875119894+119904

1198611+ 1198621119876119894+119904

1198631

+ 1198602119875119894+119904

1198612+ 1198622119876119894+119904

1198632

+ 1198603119875119867

119894+1199041198613+ 1198623119876119867

119894+1199041198633

+ 1198604119875119879

119894+1199041198614+ 1198624119876119879

119894+1199041198634))

119867

119877119894]

]

= Retr (119877119867119894+119904

119877119894) minus

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2

1003817100381710038171003817119875119894+1199041003817100381710038171003817

2

+1003817100381710038171003817119876119894+119904

1003817100381710038171003817

2

times tr (119875119867119894+119904

119860119867

1119877119894119861119867

1+ 119876119867

119894+119904119862119867

1119877119894119863119867

1

+ 119875119894+119904

119867

119860119867

2119877119894119861119867

2+ 119876119894+119904

119867

119862119867

2119877119894119863119867

2

+ 119875119894+119904

119860119867

3119877119894119861119867

3+ 119876119894+119904

119862119867

3119877119894119863119867

3

+ 119875119894+119904

119860119867

4119877119894119861119867

4+ 119876119894+119904

119862119867

4119877119894119863119867

4)

Re tr (119877119867119894+119904+1

119877119894)

= minus

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2

1003817100381710038171003817119875119894+1199041003817100381710038171003817

2

+1003817100381710038171003817119876119894+119904

1003817100381710038171003817

2

times Retr(119875119867119894+119904

119860119867

1119877119894119861119867

1+ 119876119867

119894+119904119862119867

1119877119894119863119867

1

+ 119875119894+119904

119867

119860119867

2119877119894119861119867

2+ 119876119894+119904

119867

119862119867

2119877119894119863119867

2

+ 1198613119877119867

1198941198603119875119867

119894+119904+ 1198633119877119867

1198941198623119876119867

119894+119904

+ 1198614119877119879

1198941198604119875119867

119894+119904+ 1198634119877119879

1198941198624119876119867

119894+119904)

Journal of Discrete Mathematics 11

= minus

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2

1003817100381710038171003817119875119894+1199041003817100381710038171003817

2

+1003817100381710038171003817119876119894+119904

1003817100381710038171003817

2

times Re tr [119875119867119894+119904

(119860119867

1119877119894119861119867

1+ 119860119867

2119877119894119861119867

2

+ 1198613119877119867

1198941198603+ 1198614119877119879

1198941198604)

+ 119876119867

119894+119904(119862119867

1119877119894119863119867

1+ 119862119867

2119877119894119863119867

2

+ 1198633119877119867

1198941198623+ 1198634119877119879

1198941198624)]

= minus

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2

1003817100381710038171003817119875119894+1199041003817100381710038171003817

2

+1003817100381710038171003817119876119894+119904

1003817100381710038171003817

2

times Re tr[119875119867119894+119904

(119875119894minus

10038171003817100381710038171198771198941003817100381710038171003817

2

1003817100381710038171003817119877119894minus11003817100381710038171003817

2119875119894minus1

)

+ 119876119867

119894+119904(119876119894minus

10038171003817100381710038171198771198941003817100381710038171003817

2

1003817100381710038171003817119877119894minus11003817100381710038171003817

2119876119894minus1

)]

=

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2

1003817100381710038171003817119875119894+1199041003817100381710038171003817

2

+1003817100381710038171003817119876119894+119904

1003817100381710038171003817

2

10038171003817100381710038171198771198941003817100381710038171003817

2

1003817100381710038171003817119877119894minus11003817100381710038171003817

2

times Re tr (119875119867119894+119904

119875119894minus1

+ 119876119867

119894+119904119876119894minus1

)

(29)

Repeating (28) and (29) one can easily obtain for certain 120572

and 120573

tr (119875119867119894+119904+1

119875119894+ 119876119867

119894+119904+1119876119894) = 120572 [tr (119875119867

119904+11198751+ 119876119867

119904+11198761)]

tr (119877119867119894+119904+1

119877119894) = 120573 [tr (119877119867

119904+11198771)]

(30)

Combining these two relations with (26) implies that (24)holds for 119897 = 119904 + 1 From Steps 1 and 2 the conclusion holdsby the principle of inductionWith the above two lemmas wehave the following theorem

Theorem 6 (see [32]) If the matrix equation (1) is consistentthen a solution can be obtained within finite iteration steps byusing Algorithm 3 for any initial matrices 119881

11198821

4 Numerical Example

In this section we present numerical example to illustrate theapplication of our proposed methods

Example 7 In this example we illustrate our theoreticalresults of Algorithm 3 for solving the system of matrixequation

11986011198811198611+ 11986211198821198631+ 11986021198811198612+ 11986221198821198632+ 11986031198811198671198613

+ 11986231198821198671198633+ 11986041198811198791198614+ 11986241198821198791198634= 119864

(31)

Because of the influence of the error of calculation theresidual 119877(119896) is usually unequal to zero in this process ofthe iteration We regard the matrix 119877(119896) as a zero matrix if119877(119896) lt 10

minus10Given

1198601= [

2 + 3119894 minus119894 1 + 119894

5 1 + 2119894 minus3]

1198602= [

2 + 3119894 minus119894 1 + 119894

5 1 + 2119894 minus3]

1198603= [

0 2 minus 119894 119894

minus1 + 3119894 2 0] 119860

4=[

0 1 minus 3119894 1 + 119894

0 4 + 119894 minus3119894]

1198621= [

1 + 2119894 3 minus 119894 4

minus119894 2119894 minus3] 119862

2=[

3 + 2119894 0 1 + 119894

0 4119894 1 minus 2119894]

1198623= [

1 minus 3119894 2119894 minus3119894

1 2 + 3119894 4119894] 119862

4=[

1 minus 2119894 0 2

3 minus 119894 1 + 119894 minus1]

1198611= [

[

4 + 119894 minus119894

0 1 minus 119894

4119894 2 + 2119894

]

]

1198612= [

[

0 119894

1 + 119894 0

minus1 minus 119894 3119894

]

]

1198613= [

[

0 1

minus3119894 4 + 119894

5 1 + 2119894

]

]

1198614=[

[

3 + 119894 minus1 minus 119894

0 2 minus 119894

minus1 + 119894 2

]

]

1198631= [

[

0 0

1 minus 3119894 minus119894

2119894 minus3119894

]

]

1198632= [

[

0 119894

1 + 119894 0

minus1 minus 119894 3119894

]

]

1198633= [

[

0 1

minus3119894 4 + 119894

5 1 + 2119894

]

]

1198634= [

[

3119894 minus2 + 119894

0 119894

minus2119894 minus4119894

]

]

119864 = [42 + 55119894 115 + 25119894

minus38 minus 119894 132 + 44119894]

(32)

Taking 1198811= [0 0

0 0

0 0

] and 1198821= [0 0

0 0] we apply Algorithm 3

to compute 119881119896119882119896

And iterating 42 steps we get

119881

=[

[

00126 + 18415119894 00827 + 06381119894 11221 minus 09428119894

minus06903 + 10185119894 18818 + 10203119894 09208 + 04569119894

05344 minus 04909119894 09280 + 07169119894 04872 minus 02734119894

]

]

119882

=[

[

04218 minus 09710119894 01763 + 05183119894 11331 minus 00432119894

06273 minus 01216119894 minus03902 + 04313119894 06240 + 09828119894

minus07011 minus 03418119894 03695 + 17627119894 minus03032 + 07073119894

]

]

(33)

which satisfy the matrix equation

11986011198811198611+ 11986211198821198631+ 11986021198811198612+ 11986221198821198632+ 11986031198811198671198613

+ 11986231198821198671198633+ 11986041198811198791198614+ 11986241198821198791198634= 119864

(34)

12 Journal of Discrete Mathematics

log10 rk

4

2

0

minus2

minus4

minus6

minus8

minus100 5 10 15 20 25 30 35 40 45

k (iteration number)

Figure 1The relation between the number of iterations and residualfor the example

With the corresponding residual100381710038171003817100381711987742

1003817100381710038171003817

=10038171003817100381710038171003817119864 minus 119860

1119881421198611minus 1198621119882421198631minus 1198602119881421198612minus 1198622119882421198632

minus 1198603119881119867

421198613minus 1198623119882119867

421198633minus 1198604119881119879

421198614minus 1198624119882119879

421198634

10038171003817100381710038171003817

= 66115 times 10minus11

(35)

5 Conclusions

The above Figure 1 shows the convergence curve for theresidual function 119877(119896) In this paper an iterative algorithmconstructed to solve a complex matrix equation with conju-gate and transpose of two unknowns of the form 119860

11198811198611+

11986211198821198631+11986021198811198612+11986221198821198632+11986031198811198671198613+11986231198821198671198633+11986041198811198791198614+

11986241198821198791198634

= 119864 is presented We proved that the iterativealgorithms always converge to the solution for any initialmatrices We stated and proved some lemmas and theoremswhere the solutions are obtained The proposed methodis illustrated by numerical example where the obtainednumerical results show that our technique is very neat andefficient

References

[1] F Ding P X Liu and J Ding ldquoIterative solutions of thegeneralized Sylvester matrix equations by using the hierarchicalidentification principlerdquo Applied Mathematics and Computa-tion vol 197 no 1 pp 41ndash50 2008

[2] F Ding and T Chen ldquoGradient based iterative algorithmsfor solving a class of matrix equationsrdquo IEEE Transactions onAutomatic Control vol 50 no 8 pp 1216ndash1221 2005

[3] FDing andTChen ldquoHierarchical gradient-based identificationof multivariable discrete-time systemsrdquo Automatica vol 41 no2 pp 315ndash325 2005

[4] F Ding and T Chen ldquoIterative least-squares solutions of cou-pled Sylvester matrix equationsrdquo Systems and Control Lettersvol 54 no 2 pp 95ndash107 2005

[5] F Ding and T Chen ldquoOn iterative solutions of general coupledmatrix equationsrdquo SIAM Journal on Control and Optimizationvol 44 no 6 pp 2269ndash2284 2006

[6] F Ding and T Chen ldquoHierarchical least squares identificationmethods for multivariable systemsrdquo IEEE Transactions onAutomatic Control vol 50 no 3 pp 397ndash402 2005

[7] F Ding and T Chen ldquoPerformance analysis ofmulti-innovationgradient type identification methodsrdquo Automatica vol 43 no1 pp 1ndash14 2007

[8] AGWuX ZengG RDuan andW JWu ldquoIterative solutionsto the extended Sylvester-conjugate matrix equationsrdquo AppliedMathematics and Computation vol 217 no 1 pp 130ndash142 2010

[9] B Zhou G R Duan and Z Y Li ldquoGradient based iterativealgorithm for solving coupled matrix equationsrdquo Systems andControl Letters vol 58 no 5 pp 327ndash333 2009

[10] J H Bevis F J Hall and R E Hartwing ldquoConsimilarity andthe matrix equation119860119883minus119883119861 = 119862rdquo in Current Trends inMatrixTheory pp 51ndash64 North-Holland New York NY USA 1987

[11] R A Horn and C R Johnson Matrix Analysis CambridgeUniversity Press Cambridge UK 1990

[12] H Liping ldquoConsimilarity of quaternion matrices and complexmatricesrdquo Linear Algebra and Its Applications vol 331 no 1ndash3pp 21ndash30 2001

[13] T Jiang X Cheng and L Chen ldquoAn algebraic relation betweenconsimilarity and similarity of complex matrices and its appli-cationsrdquo Journal of Physics A vol 39 no 29 pp 9215ndash92222006

[14] J H Bevis F J Hall and R E Hartwig ldquoThe matrix equation119860119883 minus 119883119861 = 119862 and its special casesrdquo SIAM Journal on MatrixAnalysis and Applications vol 9 no 3 pp 348ndash359 1988

[15] A G Wu G R Duan and H H Yu ldquoOn solutions of thematrix equations 119883119865 minus 119860119883 = 119862 and 119883119865 minus 119860119883 = 119862rdquo AppliedMathematics and Computation vol 183 no 2 pp 932ndash9412006

[16] A Navarra P L Odell and D M Young ldquoRepresentationof the general common solution to the matrix equations11986011198831198611= 1198621 11986021198831198612= 1198622with applicationsrdquo Computers and

Mathematics with Applications vol 41 no 7-8 pp 929ndash9352001

[17] J W Van der Woude ldquoOn the existence of a common solutionX to the matrix equations 119860

119894119883119861119895= 119862 (i j) E Irdquo Linear Algebra

and Its Applications vol 375 no 1ndash3 pp 135ndash145 2003[18] P Bhimasankaram ldquoCommon solutions to the linear matrix

equations 119860119883 = 119862119883119861 = 119863 and EXF = Grdquo Sankhya Series Avol 38 pp 404ndash409 1976

[19] S Kumar Mitra ldquoThe matrix equations 119860119883 = 119862119883119861 = 119863rdquoLinear Algebra and Its Applications vol 59 pp 171ndash181 1984

[20] S K Mitra ldquoA pair of simultaneous linear matrix equations11986011198831198611

= 1198621 11986021198831198612

= 1198622and a matrix programming

problemrdquo Linear Algebra and Its Applications vol 131 pp 107ndash123 1990

[21] M A Ramadan M A Abdel Naby and A M E BayoumildquoOn the explicit solutions of forms of the Sylvester and theYakubovich matrix equationsrdquo Mathematical and ComputerModelling vol 50 no 9-10 pp 1400ndash1408 2009

[22] K W E Chu ldquoSingular value and generalized singular valuedecompositions and the solution of linear matrix equationsrdquoLinear Algebra and Its Applications vol 88-89 pp 83ndash98 1987

Journal of Discrete Mathematics 13

[23] Y X Yuan ldquoThe optimal solution of linear matrix equationby matrix decompositions Mathrdquo Numerica Sinica vol 24 pp165ndash176 2002

[24] A P Liao and Y Lei ldquoLeast-squares solution with the mini-mum-norm for the matrix equation (119860119883119861 119866119883119867) = (119862119863)rdquoComputers and Mathematics with Applications vol 50 no 3-4pp 539ndash549 2005

[25] Y H Liu ldquoRanks of least squares solutions of the matrix equa-tion 119860119883119861 = 119862rdquo Computers and Mathematics with Applicationsvol 55 no 6 pp 1270ndash1278 2008

[26] X Sheng and G Chen ldquoA finite iterative method for solving apair of linear matrix equations (119860119883119861 119862119883119863) = (119864 119865)rdquo AppliedMathematics and Computation vol 189 no 2 pp 1350ndash13582007

[27] A G Wu W Liu and G R Duan ldquoOn the conjugate productof complex polynomial matricesrdquo Mathematical and ComputerModelling vol 53 no 9-10 pp 2031ndash2043 2011

[28] A GWu G FengW Liu and G R Duan ldquoThe complete solu-tion to the Sylvester-polynomial-conjugate matrix equationsrdquoMathematical and Computer Modelling vol 53 no 9-10 pp2044ndash2056 2011

[29] A G Wu B Li Y Zhang and G R Duan ldquoFinite iterativesolutions to coupled Sylvester-conjugate matrix equationsrdquoApplied Mathematical Modelling vol 35 no 3 pp 1065ndash10802011

[30] AGWu L Lv andGRDuan ldquoIterative algorithms for solvinga class of complex conjugate and transpose matrix equationsrdquoApplied Mathematics and Computation vol 217 no 21 pp8343ndash8353 2011

[31] X ZhangMatrix Analysis and Application TsinghuaUniversityPress Beijing China 2004

[32] A G Wu L Lv and M Z Hou ldquoFinite iterative algorithms forextended Sylvester-conjugate matrix equationsrdquo Mathematicaland Computer Modelling vol 54 pp 2363ndash2384 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Journal of Discrete Mathematics

= Re tr [1198771198671(1198601119881lowast1198611+ 1198621119882lowast1198631+ 1198602119881lowast1198612

+ 1198622119882lowast1198632+ 1198603119881lowast119867

1198613+ 1198623119882lowast119867

1198633

+ 1198604119881lowast119879

1198614+ 1198624119882lowast119879

1198634minus 119860111988111198611

minus 119862111988211198631minus 119860211988111198612minus 119862211988211198632

minus 1198603119881119867

11198613minus 1198623119882119867

11198633

minus 1198604119881119879

11198614minus 1198624119882119879

11198634)]

(10)

In view that 119881lowast and 119882lowast are solutions of matrix equation (1)

with this relation we have

Re tr [1198751198671

(119881lowastminus 1198811) + 119876119867

1(119882lowastminus 1198821)]

= Re tr [1198771198671(119864 minus 119860

111988111198611minus 119862111988211198631

minus 119860211988111198612

minus 119862211988211198632

minus 1198603119881119867

11198613

minus 1198623119882119867

11198633

minus 1198604119881119879

11198614minus 1198624119882119879

11198634)]

= Re tr [11987711986711198771] =

100381710038171003817100381711987711003817100381710038171003817

2

(11)

This implies that (8) holds for 119894 = 1Assume that (8) holds for = 119896 That is

Re tr [119875119867119896

(119881lowastminus 119881119896) + 119876119867

119896(119882lowastminus 119882119896)] =

10038171003817100381710038171198771198961003817100381710038171003817

2

(12)

Then we have to prove that the conclusion holds for 119894 = 119896+ 1it follows from Algorithm 3 that

Re tr [119875119867119896+1

(119881lowastminus 119881119896+1

) + 119876119867

119896+1(119882lowastminus 119882119896+1

)]

= Re

tr[

[

(119860119867

1119877119896+1

119861119867

1+ 119860119867

2119877119896+1

119861119867

2

+ 1198613119877119867

119896+11198603+ 1198614119877119879

119896+11198604

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2119875119896)

119867

(119881lowastminus 119881119896+1

)

+ (119862119867

1119877119896+1

119863119867

1+ 119862119867

2119877119896+1

119863119867

2

+ 1198633119877119867

119896+11198623+ 1198634119877119879

119896+11198624

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2119876119896)

119867

(119882lowastminus 119882119896+1

)]

]

= Retr[119877119867119896+1

1198601(119881lowastminus 119881119896+1

) 1198611

+ 119877119879

119896+11198602(119881lowastminus 119881119896+1

) 1198612

+ 119877119896+1

119861119867

3(119881lowastminus 119881119896+1

) 119860119867

3

+ 119877119896+1

1198614

119867

(119881lowastminus 119881119896+1

) 1198604

119867

+ 119877119867

119896+11198621(119882lowastminus 119882119896+1

)1198631

+ 119877119879

119896+11198622(119882lowastminus 119882119896+1

)1198632

+ 119877119896+1

119863119867

3(119882lowastminus 119882119896+1

) 119862119867

3

+ 119877119896+1

1198634

119867

(119882lowastminus 119882119896+1

) 1198624

119867

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2(119875119867

119896(119881lowastminus 119881119896+1

)

+ 119876119867

119896(119882lowastminus 119882119896+1

))]

(13)

From properties of trace and conjugate we get

Re tr [119875119867119896+1

(119881lowastminus 119881119896+1

) + 119876119867

119896+1(119882lowastminus 119882119896+1

)]

= Retr[119877119867119896+1

1198601(119881lowastminus 119881119896+1

) 1198611

+ 119877119879

119896+11198602(119881lowast minus 119881

119896+1) 1198612

+ 1198603(119881lowastminus 119881119896+1

)119867

1198613119877119867

119896+1

+ 1198604(119881lowastminus 119881119896+1

)119879

1198614119877119867

119896+1

+ 119877119867

119896+11198621(119882lowastminus 119882119896+1

)1198631

+ 119877119879

119896+11198622(119882lowast minus 119882

119896+1)1198632

+ 1198623(119882lowastminus 119882119896+1

)119867

1198633119877119867

119896+1

+ 1198624(119882lowastminus 119882119896+1

)119879

1198634119877119867

119896+1+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2

times (119875119867

119896(119881lowastminus 119881119896+1

)

+ 119876119867

119896(119882lowastminus 119882119896+1

))]

Journal of Discrete Mathematics 5

= Re tr [119877119867119896+1

1198601(119881lowastminus 119881119896+1

) 1198611

+ 119877119867

119896+11198602(119881lowast minus 119881

119896+1)1198612

+ 119877119867

119896+11198603(119881lowastminus 119881119896+1

)119867

1198613

+ 119877119867

119896+11198604(119881lowastminus 119881119896+1

)119879

1198614

+ 119877119867

119896+11198621(119882lowastminus 119882119896+1

)1198631

+ 119877119867

119896+11198622(119882lowast minus 119882

119896+1)1198632

+ 119877119867

119896+11198623(119882lowastminus 119882119896+1

)119867

1198633

+ 119877119867

119896+11198624(119882lowastminus 119882119896+1

)119879

1198634] +

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2

times Retr[119875119867119896

(119881lowastminus 119881119896minus

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2119875119896)

+ 119876119867

119896(119882lowastminus 119882119896

minus

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2119876119896)]

= Re tr [119877119867119896+1

(1198601119881lowast1198611+ 1198621119882lowast1198631

+ 1198602119881lowast1198612+ 1198622119882lowast1198632

+ 1198603119881lowast119867

1198613+ 1198623119882lowast119867

1198633

+ 1198604119881lowast119879

1198614+ 1198624119882lowast119879

1198634

minus 1198601119881119896+1

1198611minus 1198621119882119896+1

1198631minus 1198602119881119896+1

1198612

minus 1198622119882119896+1

1198632

minus 1198603119881119867

119896+11198613

minus 1198623119882119867

119896+11198633minus 1198604119881119879

119896+11198614

minus 1198624119882119879

119896+11198634)]

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2Retr[119875119867

119896(119881lowastminus 119881119896) + 119876

119867

119896(119882lowastminus 119882119896)

minus

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2(119875119867

119896119875119896+ 119876119867

119896119876119896)]

(14)

In view that 119881lowast and 119882lowast are solutions of matrix equation (1)

with relation (14) one has

Re tr [119875119867119896+1

(119881lowastminus 119881119896+1

) + 119876119867

119896+1(119882lowastminus 119882119896+1

)]

= Re tr [119877119867119896+1

(119864 minus 1198601119881119896+1

1198611minus 1198621119882119896+1

1198631

minus 1198602119881119896+1

1198612minus 1198622119882119896+1

1198632

minus 1198603119881119867

119896+11198613minus 1198623119882119867

119896+11198633

minus 1198604119881119879

119896+11198614minus 1198624119882119879

119896+11198634)]

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2[1003817100381710038171003817119877119896

1003817100381710038171003817

2

minus

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2(1003817100381710038171003817119875119896

1003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

)]

= Re tr [119877119867119896+1

119877119896+1

] =1003817100381710038171003817119877119896+1

1003817100381710038171003817

2

(15)

Then relation (8) holds by mathematical induction

Lemma 5 Suppose that the matrix equation (1) is consis-tent and the sequences 119877

119894 119875119894 and 119876

119894 are generated

by Algorithm 3 with any initial matrices 1198811 1198821 such that

119877119894

= 0 for all 119894 = 1 2 119896 and then

Re tr (119877119867119895119877119894) = 0

Re tr (119875119867119895119875119894+ 119876119867

119895119876119894) = 0 119894 119895 = 1 2 119896 119894 = 119895

(16)

Proof We apply mathematical inductionStep 1 We prove that

Re tr (119877119867119894+1

119877119894) = 0 (17)

Re tr (119875119867119894+1

119875119894+ 119876119867

119894+1119876119894) = 0 119894 = 1 2 119896 (18)

First from Algorithm 3 we have

119877119896+1

= 119864 minus 1198601119881119896+1

1198611minus 1198621119882119896+1

1198631minus 1198602119881119896+1

1198612

minus 1198622119882119896+1

1198632minus 1198603119881119867

119896+11198613

minus 1198623119882119867

119896+11198633minus 1198604119881119879

119896+11198614minus 1198624119882119879

119896+11198634

119877119896+1

= 119864 minus 1198601(119881119896+

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2119875119896)1198611

minus 1198621(119882119896+

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2119876119896)1198631

minus 1198602(119881119896+

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2119875119896)1198612

minus 1198622(119882119896+

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2119876119896)1198632

minus 1198603(119881119896+

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2119875119896)

119867

1198613

minus 1198623(119882119896+

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2119876119896)

119867

1198633

minus 1198604(119881119896+

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2119875119896)

119879

1198614

6 Journal of Discrete Mathematics

minus 1198624(119882119896+

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2119876119896)

119879

1198634

= 119864 minus 11986011198811198961198611minus 11986211198821198961198631minus 11986021198811198961198612minus 11986221198821198961198632

minus 1198603119881119867

1198961198613minus 1198623119882119867

1198961198633minus 1198604119881119879

1198961198614

minus 1198624119882119879

1198961198634minus

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

times (11986011198751198961198611+ 11986211198761198961198631+ 11986021198751198961198612

+ 11986221198761198961198632+ 1198603119875119867

1198961198613

+ 1198623119876119867

1198961198633+ 1198604119875119879

1198961198614+ 1198624119876119879

1198961198634)

119877119896+1

= 119877119896minus

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

times (11986011198751198961198611+ 11986211198761198961198631+ 11986021198751198961198612

+ 11986221198761198961198632+ 1198603119875119867

1198961198613

+ 1198623119876119867

1198961198633+ 1198604119875119879

1198961198614+ 1198624119876119879

1198961198634)

(19)

For 119894 = 1 it follows from (19) that

Re tr (11987711986721198771)

= Re

tr[

[

(1198771minus

100381710038171003817100381711987711003817100381710038171003817

2

100381710038171003817100381711987511003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2

times (119860111987511198611+ 119862111987611198631

+ 119860211987511198612+ 119862211987611198632

+ 1198603119875119867

11198613+ 1198623119876119867

11198633

+ 1198604119875119879

11198614+ 1198624119876119879

11198634))

119867

1198771]

]

= Retr (11987711986711198771) minus

100381710038171003817100381711987711003817100381710038171003817

2

100381710038171003817100381711987511003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2

times tr (1198751198671119860119867

11198771119861119867

1+ 119876119867

1119862119867

11198771119863119867

1

+ 1198751

119867

119860119867

21198771119861119867

2+ 1198761

119867

119862119867

21198771119863119867

2

+ 1198751119860119867

31198771119861119867

3+ 1198761119862119867

31198771119863119867

3

+ 1198751119860119867

41198771119861119867

4+ 1198761119862119867

41198771119863119867

4)

=10038171003817100381710038171198771

1003817100381710038171003817

2

minus

100381710038171003817100381711987711003817100381710038171003817

2

100381710038171003817100381711987511003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2

times Re tr [1198751198671119860119867

11198771119861119867

1+ 119876119867

1119862119867

11198771119863119867

1

+ 1198751

119867

119860119867

21198771119861119867

2+ 1198761

119867

119862119867

21198771119863119867

2

+ 1198613119877119867

11198603119875119867

1+ 1198633119877119867

11198623119876119867

1

+ 1198614119877119879

11198604119875119867

1+ 1198634119877119879

11198624119876119867

1]

=10038171003817100381710038171198771

1003817100381710038171003817

2

minus

100381710038171003817100381711987711003817100381710038171003817

2

100381710038171003817100381711987511003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2

times Re tr [1198751198671

(119860119867

11198771119861119867

1+ 119860119867

21198771119861119867

2

+ 1198613119877119867

11198603+ 1198614119877119879

11198604)

+ 119876119867

1(119862119867

11198771119863119867

1+ 119862119867

21198771119863119867

2

+ 1198633119877119867

11198623+ 1198634119877119879

11198624)]

=10038171003817100381710038171198771

1003817100381710038171003817

2

minus

100381710038171003817100381711987711003817100381710038171003817

2

100381710038171003817100381711987511003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2Re tr (119875119867

11198751+ 119876119867

11198761)

Re tr (11987711986721198771)

=10038171003817100381710038171198771

1003817100381710038171003817

2

minus

100381710038171003817100381711987711003817100381710038171003817

2

100381710038171003817100381711987511003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2(10038171003817100381710038171198751

1003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2

) = 0

(20)

This implies that (17) is satisfied for 119894 = 1From Algorithm 3 we have

Re tr (11987511986721198751+ 119876119867

21198761)

= Re

tr[

[

(119860119867

11198772119861119867

1+ 119860119867

21198772119861119867

2+ 1198613119877119867

21198603

+ 1198614119877119879

21198604+

100381710038171003817100381711987721003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

21198751)

119867

1198751

+ (119862119867

11198772119863119867

1+ 119862119867

21198772119863119867

2+ 1198633119877119867

21198623

+1198634119877119879

21198624+

100381710038171003817100381711987721003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

21198761)

119867

1198761]

]

Journal of Discrete Mathematics 7

= Retr[1198771198672119860111987511198611+ 1198772

119867

119860211987511198612+ 1198772119861119867

31198751119860119867

3

+ 11987721198614

119867

11987511198604

119867

+

100381710038171003817100381711987721003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

2119875119867

11198751+ 119877119867

2119862111987611198631

+ 1198772

119867

119862211987611198632+ 1198772119863119867

31198761119862119867

3

+ 11987721198634

119867

11987611198624

119867

+

100381710038171003817100381711987721003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

2119876119867

11198761]

= Retr[1198771198672119860111987511198611+ 1198772

119867

119860211987511198612+ 1198603119875119867

11198613119877119867

2

+ 1198604119875119879

11198614119877119867

2+ 119877119867

2119862111987611198631

+ 1198772

119867

119862211987611198632+ 1198623119876119867

11198633119877119867

2

+ 1198624119876119879

11198634119877119867

2+

100381710038171003817100381711987721003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

2(119875119867

11198751+ 119876119867

11198761)]

= Retr[1198771198672119860111987511198611+ 119877119867

2119860211987511198612+ 119877119867

21198603119875119867

11198613

+ 119877119867

21198604119875119879

11198614+ 119877119867

2119862111987611198631

+ 119877119867

2119862211987611198632+ 119877119867

21198623119876119867

11198633

+ 119877119867

21198624119876119879

11198634+

100381710038171003817100381711987721003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

2(119875119867

11198751+ 119876119867

11198761)]

= Retr[1198771198672(119860111987511198611+ 119862111987611198631+ 119860211987511198612

+ 119862211987611198632+ 1198603119875119867

11198613+ 1198623119876119867

11198633

+1198604119875119879

11198614+ 1198624119876119867

11198634) ]

+

100381710038171003817100381711987721003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

2(10038171003817100381710038171198751

1003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2

)

=

100381710038171003817100381711987511003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

2Re tr [119877119867

2(1198771minus 1198772)]

+

100381710038171003817100381711987721003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

2(10038171003817100381710038171198751

1003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2

)

= minus

100381710038171003817100381711987511003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

2(10038171003817100381710038171198772

1003817100381710038171003817

2

)

+

100381710038171003817100381711987721003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

2(10038171003817100381710038171198751

1003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2

) = 0

(21)This implies that (18) is satisfied for 119894 = 1

Assume that (17) and (18) hold for 119894 = 119896 minus 1 fromAlgorithm 3 we have

Re tr (119877119867119896+1

119877119896)

= Re

tr[

[

(119877119896minus

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

times (11986011198751198961198611+ 11986211198761198961198631+ 11986021198751198961198612

+ 11986221198761198961198632+ 1198603119875119867

1198961198613+ 1198623119876119867

1198961198633

+ 1198604119875119879

1198961198614+ 1198624119876119879

1198961198634))

119867

119877119896]

]

Re tr (119877119867119896+1

119877119896)

= Retr (119877119867119896119877119896) minus

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

times tr (119875119867119896119860119867

1119877119896119861119867

1+ 119876119867

119896119862119867

1119877119896119863119867

1

+ 119875119896

119867

119860119867

2119877119896119861119867

2+ 119876119896

119867

119862119867

2119877119896119863119867

2

+ 119875119896119860119867

3119877119896119861119867

3+ 119876119896119862119867

3119877119896119863119867

3

+ 119875119896119860119867

4119877119896119861119867

4+ 119876119896119862119867

4119877119896119863119867

4)

=1003817100381710038171003817119877119896

1003817100381710038171003817

2

minus

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

times Re tr (119875119867119896119860119867

1119877119896119861119867

1+ 119876119867

119896119862119867

1119877119896119863119867

1

+ 119875119896

119867

119860119867

2119877119896119861119867

2+ 119876119896

119867

119862119867

2119877119896119863119867

2

+ 1198613119877119867

1198961198603119875119867

119896+ 1198633119877119867

1198961198623119876119867

119896

+ 1198614119877119879

1198961198604119875119867

119896+ 1198634119877119879

1198961198624119876119867

119896)

=10038171003817100381710038171198771

1003817100381710038171003817

2

minus

100381710038171003817100381711987711003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

times Re tr [119875119867119896

(119860119867

1119877119896119861119867

1+ 119860119867

2119877119896119861119867

2

+ 1198613119877119867

1198961198603+ 1198614119877119879

1198961198604)

+ 119876119867

119896(119862119867

1119877119896119863119867

1+ 119862119867

2119877119896119863119867

2

+ 1198633119877119867

1198961198623+ 1198634119877119879

1198961198624)]

8 Journal of Discrete Mathematics

=1003817100381710038171003817119877119896

1003817100381710038171003817

2

minus

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

times Retr[119875119867119896

(119875119896minus

10038171003817100381710038171198771198961003817100381710038171003817

2

1003817100381710038171003817119877119896minus11003817100381710038171003817

2119875119896minus1

)

+ 119876119867

119896(119876119896minus

10038171003817100381710038171198771198961003817100381710038171003817

2

1003817100381710038171003817119877119896minus11003817100381710038171003817

2119876119896minus1

)]

=1003817100381710038171003817119877119896

1003817100381710038171003817

2

minus

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2(1003817100381710038171003817119875119896

1003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

) = 0

(22)

Thus (17) holds for 119894 = 119896Also from Algorithm 3 we have

Re tr (119875119867119896+1

119875119896+ 119876119867

119896+1119876119896)

= Re

tr [

[

(119860119867

1119877119896+1

119861119867

1+ 119860119867

2119877119896+1

119861119867

2

+ 1198613119877119867

119896+11198603+ 1198614119877119879

119896+11198604

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2119875119896)

119867

119875119896

+ (119862119867

1119877119896+1

119863119867

1+ 119862119867

2119877119896+1

119863119867

2

+ 1198633119877119867

119896+11198623+ 1198634119877119879

119896+11198624

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2119876119896)

119867

119876119896]

]

= Retr[119877119867119896+1

11986011198751198961198611+ 119877119896+1

119867

11986021198751198961198612

+ 119877119896+1

119861119867

3119875119896119860119867

3+ 119877119896+1

1198614

119867

1198751198961198604

119867

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2119875119867

119896119875119896+ 119877119867

119896+111986211198761198961198631

+ 119877119896+1

119867

11986221198761198961198632+ 119877119896+1

119863119867

3119876119896119862119867

3

+ 119877119896+1

1198634

119867

1198761198961198624

119867

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2119876119867

119896119876119896]

= Retr[119877119867119896+1

11986011198751198961198611+ 119877119896+1

119867

11986021198751198961198612

+ 1198603119875119867

1198961198613119877119867

119896+1+ 1198604119875119879

1198961198614119877119867

119896+1

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2119875119867

119896119875119896+ 119877119867

119896+111986211198761198961198631

+ 119877119896+1

119867

11986221198761198961198632+ 1198623119876119867

1198961198633119877119867

119896+1

+ 1198624119876119879

1198961198634119877119867

119896+1+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2119876119867

119896119876119896]

= Retr[119877119867119896+1

(11986011198751198961198611+ 11986211198761198961198631

+ 11986021198751198961198612+ 11986221198761198961198632+ 1198603119875119867

1198961198613

+1198623119876119867

1198961198633+ 1198604119875119879

1198961198614+ 1198624119876119879

1198961198634)

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2(119875119867

119896119875119896+ 119876119867

119896119876119896)]

=

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2Re tr (119877119867

119896+1(119877119896minus 119877119896+1

))

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2(1003817100381710038171003817119875119896

1003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

)

= minus

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2(1003817100381710038171003817119877119896+1

1003817100381710038171003817

2

)

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2(1003817100381710038171003817119875119896

1003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

) = 0

(23)

This implies that (17) and (18) hold for 119894 = 119896Then relations (17) and (18) holds bymathematical induc-

tionStep 2 We want to show that

Re (tr (119877119867119894+119897119877119894)) = 0

Re (tr (119875119867119894+119897119875119894+ 119876119867

119894+119897119876119894)) = 0

(24)

holds for 119897 ge 1 We will prove this conclusion by inductionThe case of 119897 = 1 has been proven in Step 1 Now we assumethat (24) holds for 119897 le 119904 119904 ge 1 The aim is to show that

Re (tr (119877119867119894+119904+1

119877119894)) = 0

Re (tr (119875119867119894+119904+1

119875119894+ 119876119867

119894+119904+1119876119894)) = 0

(25)

First we prove the following

Re (tr (119877119867119904+1

1198770)) = 0

Re (tr (119875119867119904+1

1198750+ 119876119867

119904+11198760)) = 0

(26)

Journal of Discrete Mathematics 9

By using Algorithm 3 from (19) and induction we have

Re tr (119877119867119904+1

1198770)

= Re

tr[

[

(119877119904minus

10038171003817100381710038171198771199041003817100381710038171003817

2

10038171003817100381710038171198751199041003817100381710038171003817

2

+1003817100381710038171003817119876119904

1003817100381710038171003817

2

times (11986011198751199041198611+ 11986211198761199041198631+ 11986021198751199041198612

+ 11986221198761199041198632+ 1198603119875119867

1199041198613+ 1198623119876119867

1199041198633

+ 1198604119875119879

1199041198614+ 1198624119876119879

1199041198634))

119867

1198770]

]

= Re tr (1198771198671199041198770) minus

10038171003817100381710038171198771199041003817100381710038171003817

2

10038171003817100381710038171198751199041003817100381710038171003817

2

+1003817100381710038171003817119876119904

1003817100381710038171003817

2

times tr (119875119867119904119860119867

11198770119861119867

1+ 119876119867

119904119862119867

11198770119863119867

1

+ 119875119904

119867

119860119867

21198770119861119867

2+ 119876119904

119867

119862119867

21198770119863119867

2

+ 119875119904119860119867

31198770119861119867

3+ 119876119904119862119867

31198770119863119867

3

+ 119875119904119860119867

41198770119861119867

4+ 119876119904119862119867

41198770119863119867

4)

Re tr (119877119867119904+1

1198770)

= Re tr (1198771198671199041198770) minus

10038171003817100381710038171198771199041003817100381710038171003817

2

10038171003817100381710038171198751199041003817100381710038171003817

2

+1003817100381710038171003817119876119904

1003817100381710038171003817

2

times Re tr (119875119867119904119860119867

11198770119861119867

1+ 119876119867

119904119862119867

11198770119863119867

1

+ 119875119904

119867

119860119867

21198770119861119867

2+ 119876119904

119867

119862119867

21198770119863119867

2

+ 1198613119877119867

01198603119875119867

119904+ 1198633119877119867

01198623119876119867

119904

+ 1198614119877119879

01198604119875119867

119904+ 1198634119877119879

01198624119876119867

119904)

= minus

10038171003817100381710038171198771199041003817100381710038171003817

2

10038171003817100381710038171198751199041003817100381710038171003817

2

+1003817100381710038171003817119876119904

1003817100381710038171003817

2

times Re tr (119875119867119904

(119860119867

11198770119861119867

1+ 119860119867

21198770119861119867

2

+ 1198613119877119867

01198603+ 1198614119877119879

01198604)

+ 119876119867

119904(119862119867

11198770119863119867

1+ 119862119867

21198770119863119867

2

+1198633119877119867

01198623+ 1198634119877119879

01198624))

= minus

10038171003817100381710038171198771199041003817100381710038171003817

2

10038171003817100381710038171198751199041003817100381710038171003817

2

+1003817100381710038171003817119876119904

1003817100381710038171003817

2Re tr (119875119867

1199041198750+ 119876119867

1199041198760) = 0

Re tr (119875119867119904+1

1198750+ 119876119867

119904+11198760)

= Retr[(119860119867

1119877119904+1

119861119867

1+ 119860119867

2119877119904+1

119861119867

2

+ 1198613119877119867

119904+11198603+ 1198614119877119879

119904+11198604

+

1003817100381710038171003817119877119904+11003817100381710038171003817

2

10038171003817100381710038171198771199041003817100381710038171003817

2119875119904)

119867

1198750

+ (119862119867

1119877119904+1

119863119867

1+ 119862119867

2119877119904+1

119863119867

2

+ 1198633119877119867

119904+11198623+ 1198634119877119879

119896+11198624

+

1003817100381710038171003817119877119904+11003817100381710038171003817

2

10038171003817100381710038171198771199041003817100381710038171003817

2119876119904)

119867

1198760]

]

= Retr[119877119867119904+1

119860111987501198611+ 119877119904+1

119867

119860211987501198612

+ 119877119904+1

119861119867

31198750119860119867

3+ 119877119904+1

1198614

119867

11987501198604

119867

+

1003817100381710038171003817119877119904+11003817100381710038171003817

2

10038171003817100381710038171198771199041003817100381710038171003817

2119875119867

1199041198750+ 119877119867

119904+1119862111987601198631

+ 119877119904+1

119867

119862211987601198632+ 119877119904+1

119863119867

31198760119862119867

3

+ 119877119904+1

1198634

119867

11987601198624

119867

+

1003817100381710038171003817119877119904+11003817100381710038171003817

2

10038171003817100381710038171198771199041003817100381710038171003817

2119876119867

1199041198760]

= Retr[119877119867119904+1

119860111987501198611+ 119877119904+1

119867

119860211987501198612

+ 1198603119875119867

01198613119877119867

119904+1+ 1198604119875119879

01198614119877119867

119904+1

+

1003817100381710038171003817119877119904+11003817100381710038171003817

2

10038171003817100381710038171198771199041003817100381710038171003817

2119875119867

1199041198750+ 119877119867

119904+1119862111987601198631

+ 119877119904+1

119867

119862211987601198632+ 1198623119876119867

01198633119877119867

119904+1

+ 1198624119876119879

01198634119877119867

119904+1+

1003817100381710038171003817119877119904+11003817100381710038171003817

2

10038171003817100381710038171198771199041003817100381710038171003817

2119876119867

1199041198760]

= Retr[119877119867119904+1

(119860111987501198611+ 119862111987601198631+ 119860211987501198612

+ 119862211987601198632+ 1198603119875119867

01198613

+ 1198623119876119867

01198633+ 1198604119875119879

01198614

+ 1198624119876119879

01198634)

+

1003817100381710038171003817119877119904+11003817100381710038171003817

2

10038171003817100381710038171198771199041003817100381710038171003817

2(119875119867

1199041198750+ 119876119867

1199041198760)]

=

100381710038171003817100381711987501003817100381710038171003817

2

+10038171003817100381710038171198760

1003817100381710038171003817

2

100381710038171003817100381711987701003817100381710038171003817

2Re tr (119877119867

119904+1(1198770minus 1198771))= 0

(27)

Then (26) is holds

10 Journal of Discrete Mathematics

From Algorithm 3 we have

Re tr (119875119867119894+119904+1

119875119894+ 119876119867

119894+119904+1119876119894)

= Retr[(119860119867

1119877119894+119904+1

119861119867

1+ 119860119867

2119877119894+119904+1

119861119867

2

+ 1198613119877119867

119894+119904+11198603+ 1198614119877119879

119894+119904+11198604

+

1003817100381710038171003817119877119894+119904+11003817100381710038171003817

2

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2119875119894+119904

)

119867

119875119894

+ (119862119867

1119877119894+119904+1

119863119867

1+ 119862119867

2119877119894+119904+1

119863119867

2

+ 1198633119877119867

119894+119904+11198623+ 1198634119877119879

119894+119904+11198624

+

1003817100381710038171003817119877119894+119904+11003817100381710038171003817

2

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2119876119894+119904

)

119867

119876119894]

]

= Retr[119877119867119894+119904+1

11986011198751198941198611+ 119877119894+119904+1

119867

11986021198751198941198612

+ 119877119894+119904+1

119861119867

3119875119894119860119867

3+ 119877119894+119904+1

1198614

119867

1198751198941198604

119867

+

1003817100381710038171003817119877119894+119904+11003817100381710038171003817

2

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2119875119867

119894+119904119875119894+ 119877119867

119894+119904+111986211198761198941198631

+ 119877119894+119904+1

119867

11986221198761198941198632+ 119877119894+119904+1

119863119867

3119876119894119862119867

3

+ 119877119894+119904+1

1198634

119867

1198761198941198624

119867

+

1003817100381710038171003817119877119894+119904+11003817100381710038171003817

2

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2119876119867

119894+119904119876119894]

= Retr[119877119867119894+119904+1

11986011198751198941198611+ 119877119894+119904+1

119867

11986021198751198941198612

+ 1198603119875119867

1198941198613119877119867

119894+119904+1+ 1198604119875119879

1198941198614119877119867

119894+119904+1

+

1003817100381710038171003817119877119894+119904+11003817100381710038171003817

2

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2119875119867

119894+119904119875119894+ 119877119867

119894+119904+111986211198761198941198631

+ 119877119894+119904+1

119867

11986221198761198941198632+ 1198623119876119867

1198941198633119877119867

119894+119904+1

+ 1198624119876119879

1198961198634119877119867

119894+119904+1+

1003817100381710038171003817119877119894+119904+11003817100381710038171003817

2

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2119876119867

119894+119904119876119894]

= Retr[119877119867119894+119904+1

(11986011198751198941198611+ 11986211198761198941198631

+ 11986021198751198941198612+ 11986221198761198941198632

+ 1198603119875119867

1198941198613+ 1198623119875119867

1198941198633

+ 1198604119875119879

1198941198614+ 1198624119876119879

1198941198634)

+

1003817100381710038171003817119877119894+119904+11003817100381710038171003817

2

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2(119875119867

119894+119904119875119894+ 119876119867

119894+119904119876119894)]

=

10038171003817100381710038171198751198941003817100381710038171003817

2

+1003817100381710038171003817119876119894

1003817100381710038171003817

2

10038171003817100381710038171198771198941003817100381710038171003817

2Re tr (119877119867

119894+119904+1(119877119894minus 119877119894+1

))

=

10038171003817100381710038171198751198941003817100381710038171003817

2

+1003817100381710038171003817119876119894

1003817100381710038171003817

2

10038171003817100381710038171198771198941003817100381710038171003817

2Re tr (119877119867

119894+119904+1119877119894)

(28)

Also from (19) we have

Re tr (119877119867119894+119904+1

119877119894)

= Re

tr[

[

(119877119894+119904

minus

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2

1003817100381710038171003817119875119894+1199041003817100381710038171003817

2

+1003817100381710038171003817119876119894+119904

1003817100381710038171003817

2

times (1198601119875119894+119904

1198611+ 1198621119876119894+119904

1198631

+ 1198602119875119894+119904

1198612+ 1198622119876119894+119904

1198632

+ 1198603119875119867

119894+1199041198613+ 1198623119876119867

119894+1199041198633

+ 1198604119875119879

119894+1199041198614+ 1198624119876119879

119894+1199041198634))

119867

119877119894]

]

= Retr (119877119867119894+119904

119877119894) minus

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2

1003817100381710038171003817119875119894+1199041003817100381710038171003817

2

+1003817100381710038171003817119876119894+119904

1003817100381710038171003817

2

times tr (119875119867119894+119904

119860119867

1119877119894119861119867

1+ 119876119867

119894+119904119862119867

1119877119894119863119867

1

+ 119875119894+119904

119867

119860119867

2119877119894119861119867

2+ 119876119894+119904

119867

119862119867

2119877119894119863119867

2

+ 119875119894+119904

119860119867

3119877119894119861119867

3+ 119876119894+119904

119862119867

3119877119894119863119867

3

+ 119875119894+119904

119860119867

4119877119894119861119867

4+ 119876119894+119904

119862119867

4119877119894119863119867

4)

Re tr (119877119867119894+119904+1

119877119894)

= minus

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2

1003817100381710038171003817119875119894+1199041003817100381710038171003817

2

+1003817100381710038171003817119876119894+119904

1003817100381710038171003817

2

times Retr(119875119867119894+119904

119860119867

1119877119894119861119867

1+ 119876119867

119894+119904119862119867

1119877119894119863119867

1

+ 119875119894+119904

119867

119860119867

2119877119894119861119867

2+ 119876119894+119904

119867

119862119867

2119877119894119863119867

2

+ 1198613119877119867

1198941198603119875119867

119894+119904+ 1198633119877119867

1198941198623119876119867

119894+119904

+ 1198614119877119879

1198941198604119875119867

119894+119904+ 1198634119877119879

1198941198624119876119867

119894+119904)

Journal of Discrete Mathematics 11

= minus

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2

1003817100381710038171003817119875119894+1199041003817100381710038171003817

2

+1003817100381710038171003817119876119894+119904

1003817100381710038171003817

2

times Re tr [119875119867119894+119904

(119860119867

1119877119894119861119867

1+ 119860119867

2119877119894119861119867

2

+ 1198613119877119867

1198941198603+ 1198614119877119879

1198941198604)

+ 119876119867

119894+119904(119862119867

1119877119894119863119867

1+ 119862119867

2119877119894119863119867

2

+ 1198633119877119867

1198941198623+ 1198634119877119879

1198941198624)]

= minus

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2

1003817100381710038171003817119875119894+1199041003817100381710038171003817

2

+1003817100381710038171003817119876119894+119904

1003817100381710038171003817

2

times Re tr[119875119867119894+119904

(119875119894minus

10038171003817100381710038171198771198941003817100381710038171003817

2

1003817100381710038171003817119877119894minus11003817100381710038171003817

2119875119894minus1

)

+ 119876119867

119894+119904(119876119894minus

10038171003817100381710038171198771198941003817100381710038171003817

2

1003817100381710038171003817119877119894minus11003817100381710038171003817

2119876119894minus1

)]

=

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2

1003817100381710038171003817119875119894+1199041003817100381710038171003817

2

+1003817100381710038171003817119876119894+119904

1003817100381710038171003817

2

10038171003817100381710038171198771198941003817100381710038171003817

2

1003817100381710038171003817119877119894minus11003817100381710038171003817

2

times Re tr (119875119867119894+119904

119875119894minus1

+ 119876119867

119894+119904119876119894minus1

)

(29)

Repeating (28) and (29) one can easily obtain for certain 120572

and 120573

tr (119875119867119894+119904+1

119875119894+ 119876119867

119894+119904+1119876119894) = 120572 [tr (119875119867

119904+11198751+ 119876119867

119904+11198761)]

tr (119877119867119894+119904+1

119877119894) = 120573 [tr (119877119867

119904+11198771)]

(30)

Combining these two relations with (26) implies that (24)holds for 119897 = 119904 + 1 From Steps 1 and 2 the conclusion holdsby the principle of inductionWith the above two lemmas wehave the following theorem

Theorem 6 (see [32]) If the matrix equation (1) is consistentthen a solution can be obtained within finite iteration steps byusing Algorithm 3 for any initial matrices 119881

11198821

4 Numerical Example

In this section we present numerical example to illustrate theapplication of our proposed methods

Example 7 In this example we illustrate our theoreticalresults of Algorithm 3 for solving the system of matrixequation

11986011198811198611+ 11986211198821198631+ 11986021198811198612+ 11986221198821198632+ 11986031198811198671198613

+ 11986231198821198671198633+ 11986041198811198791198614+ 11986241198821198791198634= 119864

(31)

Because of the influence of the error of calculation theresidual 119877(119896) is usually unequal to zero in this process ofthe iteration We regard the matrix 119877(119896) as a zero matrix if119877(119896) lt 10

minus10Given

1198601= [

2 + 3119894 minus119894 1 + 119894

5 1 + 2119894 minus3]

1198602= [

2 + 3119894 minus119894 1 + 119894

5 1 + 2119894 minus3]

1198603= [

0 2 minus 119894 119894

minus1 + 3119894 2 0] 119860

4=[

0 1 minus 3119894 1 + 119894

0 4 + 119894 minus3119894]

1198621= [

1 + 2119894 3 minus 119894 4

minus119894 2119894 minus3] 119862

2=[

3 + 2119894 0 1 + 119894

0 4119894 1 minus 2119894]

1198623= [

1 minus 3119894 2119894 minus3119894

1 2 + 3119894 4119894] 119862

4=[

1 minus 2119894 0 2

3 minus 119894 1 + 119894 minus1]

1198611= [

[

4 + 119894 minus119894

0 1 minus 119894

4119894 2 + 2119894

]

]

1198612= [

[

0 119894

1 + 119894 0

minus1 minus 119894 3119894

]

]

1198613= [

[

0 1

minus3119894 4 + 119894

5 1 + 2119894

]

]

1198614=[

[

3 + 119894 minus1 minus 119894

0 2 minus 119894

minus1 + 119894 2

]

]

1198631= [

[

0 0

1 minus 3119894 minus119894

2119894 minus3119894

]

]

1198632= [

[

0 119894

1 + 119894 0

minus1 minus 119894 3119894

]

]

1198633= [

[

0 1

minus3119894 4 + 119894

5 1 + 2119894

]

]

1198634= [

[

3119894 minus2 + 119894

0 119894

minus2119894 minus4119894

]

]

119864 = [42 + 55119894 115 + 25119894

minus38 minus 119894 132 + 44119894]

(32)

Taking 1198811= [0 0

0 0

0 0

] and 1198821= [0 0

0 0] we apply Algorithm 3

to compute 119881119896119882119896

And iterating 42 steps we get

119881

=[

[

00126 + 18415119894 00827 + 06381119894 11221 minus 09428119894

minus06903 + 10185119894 18818 + 10203119894 09208 + 04569119894

05344 minus 04909119894 09280 + 07169119894 04872 minus 02734119894

]

]

119882

=[

[

04218 minus 09710119894 01763 + 05183119894 11331 minus 00432119894

06273 minus 01216119894 minus03902 + 04313119894 06240 + 09828119894

minus07011 minus 03418119894 03695 + 17627119894 minus03032 + 07073119894

]

]

(33)

which satisfy the matrix equation

11986011198811198611+ 11986211198821198631+ 11986021198811198612+ 11986221198821198632+ 11986031198811198671198613

+ 11986231198821198671198633+ 11986041198811198791198614+ 11986241198821198791198634= 119864

(34)

12 Journal of Discrete Mathematics

log10 rk

4

2

0

minus2

minus4

minus6

minus8

minus100 5 10 15 20 25 30 35 40 45

k (iteration number)

Figure 1The relation between the number of iterations and residualfor the example

With the corresponding residual100381710038171003817100381711987742

1003817100381710038171003817

=10038171003817100381710038171003817119864 minus 119860

1119881421198611minus 1198621119882421198631minus 1198602119881421198612minus 1198622119882421198632

minus 1198603119881119867

421198613minus 1198623119882119867

421198633minus 1198604119881119879

421198614minus 1198624119882119879

421198634

10038171003817100381710038171003817

= 66115 times 10minus11

(35)

5 Conclusions

The above Figure 1 shows the convergence curve for theresidual function 119877(119896) In this paper an iterative algorithmconstructed to solve a complex matrix equation with conju-gate and transpose of two unknowns of the form 119860

11198811198611+

11986211198821198631+11986021198811198612+11986221198821198632+11986031198811198671198613+11986231198821198671198633+11986041198811198791198614+

11986241198821198791198634

= 119864 is presented We proved that the iterativealgorithms always converge to the solution for any initialmatrices We stated and proved some lemmas and theoremswhere the solutions are obtained The proposed methodis illustrated by numerical example where the obtainednumerical results show that our technique is very neat andefficient

References

[1] F Ding P X Liu and J Ding ldquoIterative solutions of thegeneralized Sylvester matrix equations by using the hierarchicalidentification principlerdquo Applied Mathematics and Computa-tion vol 197 no 1 pp 41ndash50 2008

[2] F Ding and T Chen ldquoGradient based iterative algorithmsfor solving a class of matrix equationsrdquo IEEE Transactions onAutomatic Control vol 50 no 8 pp 1216ndash1221 2005

[3] FDing andTChen ldquoHierarchical gradient-based identificationof multivariable discrete-time systemsrdquo Automatica vol 41 no2 pp 315ndash325 2005

[4] F Ding and T Chen ldquoIterative least-squares solutions of cou-pled Sylvester matrix equationsrdquo Systems and Control Lettersvol 54 no 2 pp 95ndash107 2005

[5] F Ding and T Chen ldquoOn iterative solutions of general coupledmatrix equationsrdquo SIAM Journal on Control and Optimizationvol 44 no 6 pp 2269ndash2284 2006

[6] F Ding and T Chen ldquoHierarchical least squares identificationmethods for multivariable systemsrdquo IEEE Transactions onAutomatic Control vol 50 no 3 pp 397ndash402 2005

[7] F Ding and T Chen ldquoPerformance analysis ofmulti-innovationgradient type identification methodsrdquo Automatica vol 43 no1 pp 1ndash14 2007

[8] AGWuX ZengG RDuan andW JWu ldquoIterative solutionsto the extended Sylvester-conjugate matrix equationsrdquo AppliedMathematics and Computation vol 217 no 1 pp 130ndash142 2010

[9] B Zhou G R Duan and Z Y Li ldquoGradient based iterativealgorithm for solving coupled matrix equationsrdquo Systems andControl Letters vol 58 no 5 pp 327ndash333 2009

[10] J H Bevis F J Hall and R E Hartwing ldquoConsimilarity andthe matrix equation119860119883minus119883119861 = 119862rdquo in Current Trends inMatrixTheory pp 51ndash64 North-Holland New York NY USA 1987

[11] R A Horn and C R Johnson Matrix Analysis CambridgeUniversity Press Cambridge UK 1990

[12] H Liping ldquoConsimilarity of quaternion matrices and complexmatricesrdquo Linear Algebra and Its Applications vol 331 no 1ndash3pp 21ndash30 2001

[13] T Jiang X Cheng and L Chen ldquoAn algebraic relation betweenconsimilarity and similarity of complex matrices and its appli-cationsrdquo Journal of Physics A vol 39 no 29 pp 9215ndash92222006

[14] J H Bevis F J Hall and R E Hartwig ldquoThe matrix equation119860119883 minus 119883119861 = 119862 and its special casesrdquo SIAM Journal on MatrixAnalysis and Applications vol 9 no 3 pp 348ndash359 1988

[15] A G Wu G R Duan and H H Yu ldquoOn solutions of thematrix equations 119883119865 minus 119860119883 = 119862 and 119883119865 minus 119860119883 = 119862rdquo AppliedMathematics and Computation vol 183 no 2 pp 932ndash9412006

[16] A Navarra P L Odell and D M Young ldquoRepresentationof the general common solution to the matrix equations11986011198831198611= 1198621 11986021198831198612= 1198622with applicationsrdquo Computers and

Mathematics with Applications vol 41 no 7-8 pp 929ndash9352001

[17] J W Van der Woude ldquoOn the existence of a common solutionX to the matrix equations 119860

119894119883119861119895= 119862 (i j) E Irdquo Linear Algebra

and Its Applications vol 375 no 1ndash3 pp 135ndash145 2003[18] P Bhimasankaram ldquoCommon solutions to the linear matrix

equations 119860119883 = 119862119883119861 = 119863 and EXF = Grdquo Sankhya Series Avol 38 pp 404ndash409 1976

[19] S Kumar Mitra ldquoThe matrix equations 119860119883 = 119862119883119861 = 119863rdquoLinear Algebra and Its Applications vol 59 pp 171ndash181 1984

[20] S K Mitra ldquoA pair of simultaneous linear matrix equations11986011198831198611

= 1198621 11986021198831198612

= 1198622and a matrix programming

problemrdquo Linear Algebra and Its Applications vol 131 pp 107ndash123 1990

[21] M A Ramadan M A Abdel Naby and A M E BayoumildquoOn the explicit solutions of forms of the Sylvester and theYakubovich matrix equationsrdquo Mathematical and ComputerModelling vol 50 no 9-10 pp 1400ndash1408 2009

[22] K W E Chu ldquoSingular value and generalized singular valuedecompositions and the solution of linear matrix equationsrdquoLinear Algebra and Its Applications vol 88-89 pp 83ndash98 1987

Journal of Discrete Mathematics 13

[23] Y X Yuan ldquoThe optimal solution of linear matrix equationby matrix decompositions Mathrdquo Numerica Sinica vol 24 pp165ndash176 2002

[24] A P Liao and Y Lei ldquoLeast-squares solution with the mini-mum-norm for the matrix equation (119860119883119861 119866119883119867) = (119862119863)rdquoComputers and Mathematics with Applications vol 50 no 3-4pp 539ndash549 2005

[25] Y H Liu ldquoRanks of least squares solutions of the matrix equa-tion 119860119883119861 = 119862rdquo Computers and Mathematics with Applicationsvol 55 no 6 pp 1270ndash1278 2008

[26] X Sheng and G Chen ldquoA finite iterative method for solving apair of linear matrix equations (119860119883119861 119862119883119863) = (119864 119865)rdquo AppliedMathematics and Computation vol 189 no 2 pp 1350ndash13582007

[27] A G Wu W Liu and G R Duan ldquoOn the conjugate productof complex polynomial matricesrdquo Mathematical and ComputerModelling vol 53 no 9-10 pp 2031ndash2043 2011

[28] A GWu G FengW Liu and G R Duan ldquoThe complete solu-tion to the Sylvester-polynomial-conjugate matrix equationsrdquoMathematical and Computer Modelling vol 53 no 9-10 pp2044ndash2056 2011

[29] A G Wu B Li Y Zhang and G R Duan ldquoFinite iterativesolutions to coupled Sylvester-conjugate matrix equationsrdquoApplied Mathematical Modelling vol 35 no 3 pp 1065ndash10802011

[30] AGWu L Lv andGRDuan ldquoIterative algorithms for solvinga class of complex conjugate and transpose matrix equationsrdquoApplied Mathematics and Computation vol 217 no 21 pp8343ndash8353 2011

[31] X ZhangMatrix Analysis and Application TsinghuaUniversityPress Beijing China 2004

[32] A G Wu L Lv and M Z Hou ldquoFinite iterative algorithms forextended Sylvester-conjugate matrix equationsrdquo Mathematicaland Computer Modelling vol 54 pp 2363ndash2384 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Discrete Mathematics 5

= Re tr [119877119867119896+1

1198601(119881lowastminus 119881119896+1

) 1198611

+ 119877119867

119896+11198602(119881lowast minus 119881

119896+1)1198612

+ 119877119867

119896+11198603(119881lowastminus 119881119896+1

)119867

1198613

+ 119877119867

119896+11198604(119881lowastminus 119881119896+1

)119879

1198614

+ 119877119867

119896+11198621(119882lowastminus 119882119896+1

)1198631

+ 119877119867

119896+11198622(119882lowast minus 119882

119896+1)1198632

+ 119877119867

119896+11198623(119882lowastminus 119882119896+1

)119867

1198633

+ 119877119867

119896+11198624(119882lowastminus 119882119896+1

)119879

1198634] +

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2

times Retr[119875119867119896

(119881lowastminus 119881119896minus

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2119875119896)

+ 119876119867

119896(119882lowastminus 119882119896

minus

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2119876119896)]

= Re tr [119877119867119896+1

(1198601119881lowast1198611+ 1198621119882lowast1198631

+ 1198602119881lowast1198612+ 1198622119882lowast1198632

+ 1198603119881lowast119867

1198613+ 1198623119882lowast119867

1198633

+ 1198604119881lowast119879

1198614+ 1198624119882lowast119879

1198634

minus 1198601119881119896+1

1198611minus 1198621119882119896+1

1198631minus 1198602119881119896+1

1198612

minus 1198622119882119896+1

1198632

minus 1198603119881119867

119896+11198613

minus 1198623119882119867

119896+11198633minus 1198604119881119879

119896+11198614

minus 1198624119882119879

119896+11198634)]

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2Retr[119875119867

119896(119881lowastminus 119881119896) + 119876

119867

119896(119882lowastminus 119882119896)

minus

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2(119875119867

119896119875119896+ 119876119867

119896119876119896)]

(14)

In view that 119881lowast and 119882lowast are solutions of matrix equation (1)

with relation (14) one has

Re tr [119875119867119896+1

(119881lowastminus 119881119896+1

) + 119876119867

119896+1(119882lowastminus 119882119896+1

)]

= Re tr [119877119867119896+1

(119864 minus 1198601119881119896+1

1198611minus 1198621119882119896+1

1198631

minus 1198602119881119896+1

1198612minus 1198622119882119896+1

1198632

minus 1198603119881119867

119896+11198613minus 1198623119882119867

119896+11198633

minus 1198604119881119879

119896+11198614minus 1198624119882119879

119896+11198634)]

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2[1003817100381710038171003817119877119896

1003817100381710038171003817

2

minus

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2(1003817100381710038171003817119875119896

1003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

)]

= Re tr [119877119867119896+1

119877119896+1

] =1003817100381710038171003817119877119896+1

1003817100381710038171003817

2

(15)

Then relation (8) holds by mathematical induction

Lemma 5 Suppose that the matrix equation (1) is consis-tent and the sequences 119877

119894 119875119894 and 119876

119894 are generated

by Algorithm 3 with any initial matrices 1198811 1198821 such that

119877119894

= 0 for all 119894 = 1 2 119896 and then

Re tr (119877119867119895119877119894) = 0

Re tr (119875119867119895119875119894+ 119876119867

119895119876119894) = 0 119894 119895 = 1 2 119896 119894 = 119895

(16)

Proof We apply mathematical inductionStep 1 We prove that

Re tr (119877119867119894+1

119877119894) = 0 (17)

Re tr (119875119867119894+1

119875119894+ 119876119867

119894+1119876119894) = 0 119894 = 1 2 119896 (18)

First from Algorithm 3 we have

119877119896+1

= 119864 minus 1198601119881119896+1

1198611minus 1198621119882119896+1

1198631minus 1198602119881119896+1

1198612

minus 1198622119882119896+1

1198632minus 1198603119881119867

119896+11198613

minus 1198623119882119867

119896+11198633minus 1198604119881119879

119896+11198614minus 1198624119882119879

119896+11198634

119877119896+1

= 119864 minus 1198601(119881119896+

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2119875119896)1198611

minus 1198621(119882119896+

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2119876119896)1198631

minus 1198602(119881119896+

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2119875119896)1198612

minus 1198622(119882119896+

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2119876119896)1198632

minus 1198603(119881119896+

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2119875119896)

119867

1198613

minus 1198623(119882119896+

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2119876119896)

119867

1198633

minus 1198604(119881119896+

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2119875119896)

119879

1198614

6 Journal of Discrete Mathematics

minus 1198624(119882119896+

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2119876119896)

119879

1198634

= 119864 minus 11986011198811198961198611minus 11986211198821198961198631minus 11986021198811198961198612minus 11986221198821198961198632

minus 1198603119881119867

1198961198613minus 1198623119882119867

1198961198633minus 1198604119881119879

1198961198614

minus 1198624119882119879

1198961198634minus

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

times (11986011198751198961198611+ 11986211198761198961198631+ 11986021198751198961198612

+ 11986221198761198961198632+ 1198603119875119867

1198961198613

+ 1198623119876119867

1198961198633+ 1198604119875119879

1198961198614+ 1198624119876119879

1198961198634)

119877119896+1

= 119877119896minus

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

times (11986011198751198961198611+ 11986211198761198961198631+ 11986021198751198961198612

+ 11986221198761198961198632+ 1198603119875119867

1198961198613

+ 1198623119876119867

1198961198633+ 1198604119875119879

1198961198614+ 1198624119876119879

1198961198634)

(19)

For 119894 = 1 it follows from (19) that

Re tr (11987711986721198771)

= Re

tr[

[

(1198771minus

100381710038171003817100381711987711003817100381710038171003817

2

100381710038171003817100381711987511003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2

times (119860111987511198611+ 119862111987611198631

+ 119860211987511198612+ 119862211987611198632

+ 1198603119875119867

11198613+ 1198623119876119867

11198633

+ 1198604119875119879

11198614+ 1198624119876119879

11198634))

119867

1198771]

]

= Retr (11987711986711198771) minus

100381710038171003817100381711987711003817100381710038171003817

2

100381710038171003817100381711987511003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2

times tr (1198751198671119860119867

11198771119861119867

1+ 119876119867

1119862119867

11198771119863119867

1

+ 1198751

119867

119860119867

21198771119861119867

2+ 1198761

119867

119862119867

21198771119863119867

2

+ 1198751119860119867

31198771119861119867

3+ 1198761119862119867

31198771119863119867

3

+ 1198751119860119867

41198771119861119867

4+ 1198761119862119867

41198771119863119867

4)

=10038171003817100381710038171198771

1003817100381710038171003817

2

minus

100381710038171003817100381711987711003817100381710038171003817

2

100381710038171003817100381711987511003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2

times Re tr [1198751198671119860119867

11198771119861119867

1+ 119876119867

1119862119867

11198771119863119867

1

+ 1198751

119867

119860119867

21198771119861119867

2+ 1198761

119867

119862119867

21198771119863119867

2

+ 1198613119877119867

11198603119875119867

1+ 1198633119877119867

11198623119876119867

1

+ 1198614119877119879

11198604119875119867

1+ 1198634119877119879

11198624119876119867

1]

=10038171003817100381710038171198771

1003817100381710038171003817

2

minus

100381710038171003817100381711987711003817100381710038171003817

2

100381710038171003817100381711987511003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2

times Re tr [1198751198671

(119860119867

11198771119861119867

1+ 119860119867

21198771119861119867

2

+ 1198613119877119867

11198603+ 1198614119877119879

11198604)

+ 119876119867

1(119862119867

11198771119863119867

1+ 119862119867

21198771119863119867

2

+ 1198633119877119867

11198623+ 1198634119877119879

11198624)]

=10038171003817100381710038171198771

1003817100381710038171003817

2

minus

100381710038171003817100381711987711003817100381710038171003817

2

100381710038171003817100381711987511003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2Re tr (119875119867

11198751+ 119876119867

11198761)

Re tr (11987711986721198771)

=10038171003817100381710038171198771

1003817100381710038171003817

2

minus

100381710038171003817100381711987711003817100381710038171003817

2

100381710038171003817100381711987511003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2(10038171003817100381710038171198751

1003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2

) = 0

(20)

This implies that (17) is satisfied for 119894 = 1From Algorithm 3 we have

Re tr (11987511986721198751+ 119876119867

21198761)

= Re

tr[

[

(119860119867

11198772119861119867

1+ 119860119867

21198772119861119867

2+ 1198613119877119867

21198603

+ 1198614119877119879

21198604+

100381710038171003817100381711987721003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

21198751)

119867

1198751

+ (119862119867

11198772119863119867

1+ 119862119867

21198772119863119867

2+ 1198633119877119867

21198623

+1198634119877119879

21198624+

100381710038171003817100381711987721003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

21198761)

119867

1198761]

]

Journal of Discrete Mathematics 7

= Retr[1198771198672119860111987511198611+ 1198772

119867

119860211987511198612+ 1198772119861119867

31198751119860119867

3

+ 11987721198614

119867

11987511198604

119867

+

100381710038171003817100381711987721003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

2119875119867

11198751+ 119877119867

2119862111987611198631

+ 1198772

119867

119862211987611198632+ 1198772119863119867

31198761119862119867

3

+ 11987721198634

119867

11987611198624

119867

+

100381710038171003817100381711987721003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

2119876119867

11198761]

= Retr[1198771198672119860111987511198611+ 1198772

119867

119860211987511198612+ 1198603119875119867

11198613119877119867

2

+ 1198604119875119879

11198614119877119867

2+ 119877119867

2119862111987611198631

+ 1198772

119867

119862211987611198632+ 1198623119876119867

11198633119877119867

2

+ 1198624119876119879

11198634119877119867

2+

100381710038171003817100381711987721003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

2(119875119867

11198751+ 119876119867

11198761)]

= Retr[1198771198672119860111987511198611+ 119877119867

2119860211987511198612+ 119877119867

21198603119875119867

11198613

+ 119877119867

21198604119875119879

11198614+ 119877119867

2119862111987611198631

+ 119877119867

2119862211987611198632+ 119877119867

21198623119876119867

11198633

+ 119877119867

21198624119876119879

11198634+

100381710038171003817100381711987721003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

2(119875119867

11198751+ 119876119867

11198761)]

= Retr[1198771198672(119860111987511198611+ 119862111987611198631+ 119860211987511198612

+ 119862211987611198632+ 1198603119875119867

11198613+ 1198623119876119867

11198633

+1198604119875119879

11198614+ 1198624119876119867

11198634) ]

+

100381710038171003817100381711987721003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

2(10038171003817100381710038171198751

1003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2

)

=

100381710038171003817100381711987511003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

2Re tr [119877119867

2(1198771minus 1198772)]

+

100381710038171003817100381711987721003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

2(10038171003817100381710038171198751

1003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2

)

= minus

100381710038171003817100381711987511003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

2(10038171003817100381710038171198772

1003817100381710038171003817

2

)

+

100381710038171003817100381711987721003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

2(10038171003817100381710038171198751

1003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2

) = 0

(21)This implies that (18) is satisfied for 119894 = 1

Assume that (17) and (18) hold for 119894 = 119896 minus 1 fromAlgorithm 3 we have

Re tr (119877119867119896+1

119877119896)

= Re

tr[

[

(119877119896minus

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

times (11986011198751198961198611+ 11986211198761198961198631+ 11986021198751198961198612

+ 11986221198761198961198632+ 1198603119875119867

1198961198613+ 1198623119876119867

1198961198633

+ 1198604119875119879

1198961198614+ 1198624119876119879

1198961198634))

119867

119877119896]

]

Re tr (119877119867119896+1

119877119896)

= Retr (119877119867119896119877119896) minus

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

times tr (119875119867119896119860119867

1119877119896119861119867

1+ 119876119867

119896119862119867

1119877119896119863119867

1

+ 119875119896

119867

119860119867

2119877119896119861119867

2+ 119876119896

119867

119862119867

2119877119896119863119867

2

+ 119875119896119860119867

3119877119896119861119867

3+ 119876119896119862119867

3119877119896119863119867

3

+ 119875119896119860119867

4119877119896119861119867

4+ 119876119896119862119867

4119877119896119863119867

4)

=1003817100381710038171003817119877119896

1003817100381710038171003817

2

minus

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

times Re tr (119875119867119896119860119867

1119877119896119861119867

1+ 119876119867

119896119862119867

1119877119896119863119867

1

+ 119875119896

119867

119860119867

2119877119896119861119867

2+ 119876119896

119867

119862119867

2119877119896119863119867

2

+ 1198613119877119867

1198961198603119875119867

119896+ 1198633119877119867

1198961198623119876119867

119896

+ 1198614119877119879

1198961198604119875119867

119896+ 1198634119877119879

1198961198624119876119867

119896)

=10038171003817100381710038171198771

1003817100381710038171003817

2

minus

100381710038171003817100381711987711003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

times Re tr [119875119867119896

(119860119867

1119877119896119861119867

1+ 119860119867

2119877119896119861119867

2

+ 1198613119877119867

1198961198603+ 1198614119877119879

1198961198604)

+ 119876119867

119896(119862119867

1119877119896119863119867

1+ 119862119867

2119877119896119863119867

2

+ 1198633119877119867

1198961198623+ 1198634119877119879

1198961198624)]

8 Journal of Discrete Mathematics

=1003817100381710038171003817119877119896

1003817100381710038171003817

2

minus

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

times Retr[119875119867119896

(119875119896minus

10038171003817100381710038171198771198961003817100381710038171003817

2

1003817100381710038171003817119877119896minus11003817100381710038171003817

2119875119896minus1

)

+ 119876119867

119896(119876119896minus

10038171003817100381710038171198771198961003817100381710038171003817

2

1003817100381710038171003817119877119896minus11003817100381710038171003817

2119876119896minus1

)]

=1003817100381710038171003817119877119896

1003817100381710038171003817

2

minus

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2(1003817100381710038171003817119875119896

1003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

) = 0

(22)

Thus (17) holds for 119894 = 119896Also from Algorithm 3 we have

Re tr (119875119867119896+1

119875119896+ 119876119867

119896+1119876119896)

= Re

tr [

[

(119860119867

1119877119896+1

119861119867

1+ 119860119867

2119877119896+1

119861119867

2

+ 1198613119877119867

119896+11198603+ 1198614119877119879

119896+11198604

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2119875119896)

119867

119875119896

+ (119862119867

1119877119896+1

119863119867

1+ 119862119867

2119877119896+1

119863119867

2

+ 1198633119877119867

119896+11198623+ 1198634119877119879

119896+11198624

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2119876119896)

119867

119876119896]

]

= Retr[119877119867119896+1

11986011198751198961198611+ 119877119896+1

119867

11986021198751198961198612

+ 119877119896+1

119861119867

3119875119896119860119867

3+ 119877119896+1

1198614

119867

1198751198961198604

119867

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2119875119867

119896119875119896+ 119877119867

119896+111986211198761198961198631

+ 119877119896+1

119867

11986221198761198961198632+ 119877119896+1

119863119867

3119876119896119862119867

3

+ 119877119896+1

1198634

119867

1198761198961198624

119867

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2119876119867

119896119876119896]

= Retr[119877119867119896+1

11986011198751198961198611+ 119877119896+1

119867

11986021198751198961198612

+ 1198603119875119867

1198961198613119877119867

119896+1+ 1198604119875119879

1198961198614119877119867

119896+1

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2119875119867

119896119875119896+ 119877119867

119896+111986211198761198961198631

+ 119877119896+1

119867

11986221198761198961198632+ 1198623119876119867

1198961198633119877119867

119896+1

+ 1198624119876119879

1198961198634119877119867

119896+1+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2119876119867

119896119876119896]

= Retr[119877119867119896+1

(11986011198751198961198611+ 11986211198761198961198631

+ 11986021198751198961198612+ 11986221198761198961198632+ 1198603119875119867

1198961198613

+1198623119876119867

1198961198633+ 1198604119875119879

1198961198614+ 1198624119876119879

1198961198634)

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2(119875119867

119896119875119896+ 119876119867

119896119876119896)]

=

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2Re tr (119877119867

119896+1(119877119896minus 119877119896+1

))

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2(1003817100381710038171003817119875119896

1003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

)

= minus

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2(1003817100381710038171003817119877119896+1

1003817100381710038171003817

2

)

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2(1003817100381710038171003817119875119896

1003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

) = 0

(23)

This implies that (17) and (18) hold for 119894 = 119896Then relations (17) and (18) holds bymathematical induc-

tionStep 2 We want to show that

Re (tr (119877119867119894+119897119877119894)) = 0

Re (tr (119875119867119894+119897119875119894+ 119876119867

119894+119897119876119894)) = 0

(24)

holds for 119897 ge 1 We will prove this conclusion by inductionThe case of 119897 = 1 has been proven in Step 1 Now we assumethat (24) holds for 119897 le 119904 119904 ge 1 The aim is to show that

Re (tr (119877119867119894+119904+1

119877119894)) = 0

Re (tr (119875119867119894+119904+1

119875119894+ 119876119867

119894+119904+1119876119894)) = 0

(25)

First we prove the following

Re (tr (119877119867119904+1

1198770)) = 0

Re (tr (119875119867119904+1

1198750+ 119876119867

119904+11198760)) = 0

(26)

Journal of Discrete Mathematics 9

By using Algorithm 3 from (19) and induction we have

Re tr (119877119867119904+1

1198770)

= Re

tr[

[

(119877119904minus

10038171003817100381710038171198771199041003817100381710038171003817

2

10038171003817100381710038171198751199041003817100381710038171003817

2

+1003817100381710038171003817119876119904

1003817100381710038171003817

2

times (11986011198751199041198611+ 11986211198761199041198631+ 11986021198751199041198612

+ 11986221198761199041198632+ 1198603119875119867

1199041198613+ 1198623119876119867

1199041198633

+ 1198604119875119879

1199041198614+ 1198624119876119879

1199041198634))

119867

1198770]

]

= Re tr (1198771198671199041198770) minus

10038171003817100381710038171198771199041003817100381710038171003817

2

10038171003817100381710038171198751199041003817100381710038171003817

2

+1003817100381710038171003817119876119904

1003817100381710038171003817

2

times tr (119875119867119904119860119867

11198770119861119867

1+ 119876119867

119904119862119867

11198770119863119867

1

+ 119875119904

119867

119860119867

21198770119861119867

2+ 119876119904

119867

119862119867

21198770119863119867

2

+ 119875119904119860119867

31198770119861119867

3+ 119876119904119862119867

31198770119863119867

3

+ 119875119904119860119867

41198770119861119867

4+ 119876119904119862119867

41198770119863119867

4)

Re tr (119877119867119904+1

1198770)

= Re tr (1198771198671199041198770) minus

10038171003817100381710038171198771199041003817100381710038171003817

2

10038171003817100381710038171198751199041003817100381710038171003817

2

+1003817100381710038171003817119876119904

1003817100381710038171003817

2

times Re tr (119875119867119904119860119867

11198770119861119867

1+ 119876119867

119904119862119867

11198770119863119867

1

+ 119875119904

119867

119860119867

21198770119861119867

2+ 119876119904

119867

119862119867

21198770119863119867

2

+ 1198613119877119867

01198603119875119867

119904+ 1198633119877119867

01198623119876119867

119904

+ 1198614119877119879

01198604119875119867

119904+ 1198634119877119879

01198624119876119867

119904)

= minus

10038171003817100381710038171198771199041003817100381710038171003817

2

10038171003817100381710038171198751199041003817100381710038171003817

2

+1003817100381710038171003817119876119904

1003817100381710038171003817

2

times Re tr (119875119867119904

(119860119867

11198770119861119867

1+ 119860119867

21198770119861119867

2

+ 1198613119877119867

01198603+ 1198614119877119879

01198604)

+ 119876119867

119904(119862119867

11198770119863119867

1+ 119862119867

21198770119863119867

2

+1198633119877119867

01198623+ 1198634119877119879

01198624))

= minus

10038171003817100381710038171198771199041003817100381710038171003817

2

10038171003817100381710038171198751199041003817100381710038171003817

2

+1003817100381710038171003817119876119904

1003817100381710038171003817

2Re tr (119875119867

1199041198750+ 119876119867

1199041198760) = 0

Re tr (119875119867119904+1

1198750+ 119876119867

119904+11198760)

= Retr[(119860119867

1119877119904+1

119861119867

1+ 119860119867

2119877119904+1

119861119867

2

+ 1198613119877119867

119904+11198603+ 1198614119877119879

119904+11198604

+

1003817100381710038171003817119877119904+11003817100381710038171003817

2

10038171003817100381710038171198771199041003817100381710038171003817

2119875119904)

119867

1198750

+ (119862119867

1119877119904+1

119863119867

1+ 119862119867

2119877119904+1

119863119867

2

+ 1198633119877119867

119904+11198623+ 1198634119877119879

119896+11198624

+

1003817100381710038171003817119877119904+11003817100381710038171003817

2

10038171003817100381710038171198771199041003817100381710038171003817

2119876119904)

119867

1198760]

]

= Retr[119877119867119904+1

119860111987501198611+ 119877119904+1

119867

119860211987501198612

+ 119877119904+1

119861119867

31198750119860119867

3+ 119877119904+1

1198614

119867

11987501198604

119867

+

1003817100381710038171003817119877119904+11003817100381710038171003817

2

10038171003817100381710038171198771199041003817100381710038171003817

2119875119867

1199041198750+ 119877119867

119904+1119862111987601198631

+ 119877119904+1

119867

119862211987601198632+ 119877119904+1

119863119867

31198760119862119867

3

+ 119877119904+1

1198634

119867

11987601198624

119867

+

1003817100381710038171003817119877119904+11003817100381710038171003817

2

10038171003817100381710038171198771199041003817100381710038171003817

2119876119867

1199041198760]

= Retr[119877119867119904+1

119860111987501198611+ 119877119904+1

119867

119860211987501198612

+ 1198603119875119867

01198613119877119867

119904+1+ 1198604119875119879

01198614119877119867

119904+1

+

1003817100381710038171003817119877119904+11003817100381710038171003817

2

10038171003817100381710038171198771199041003817100381710038171003817

2119875119867

1199041198750+ 119877119867

119904+1119862111987601198631

+ 119877119904+1

119867

119862211987601198632+ 1198623119876119867

01198633119877119867

119904+1

+ 1198624119876119879

01198634119877119867

119904+1+

1003817100381710038171003817119877119904+11003817100381710038171003817

2

10038171003817100381710038171198771199041003817100381710038171003817

2119876119867

1199041198760]

= Retr[119877119867119904+1

(119860111987501198611+ 119862111987601198631+ 119860211987501198612

+ 119862211987601198632+ 1198603119875119867

01198613

+ 1198623119876119867

01198633+ 1198604119875119879

01198614

+ 1198624119876119879

01198634)

+

1003817100381710038171003817119877119904+11003817100381710038171003817

2

10038171003817100381710038171198771199041003817100381710038171003817

2(119875119867

1199041198750+ 119876119867

1199041198760)]

=

100381710038171003817100381711987501003817100381710038171003817

2

+10038171003817100381710038171198760

1003817100381710038171003817

2

100381710038171003817100381711987701003817100381710038171003817

2Re tr (119877119867

119904+1(1198770minus 1198771))= 0

(27)

Then (26) is holds

10 Journal of Discrete Mathematics

From Algorithm 3 we have

Re tr (119875119867119894+119904+1

119875119894+ 119876119867

119894+119904+1119876119894)

= Retr[(119860119867

1119877119894+119904+1

119861119867

1+ 119860119867

2119877119894+119904+1

119861119867

2

+ 1198613119877119867

119894+119904+11198603+ 1198614119877119879

119894+119904+11198604

+

1003817100381710038171003817119877119894+119904+11003817100381710038171003817

2

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2119875119894+119904

)

119867

119875119894

+ (119862119867

1119877119894+119904+1

119863119867

1+ 119862119867

2119877119894+119904+1

119863119867

2

+ 1198633119877119867

119894+119904+11198623+ 1198634119877119879

119894+119904+11198624

+

1003817100381710038171003817119877119894+119904+11003817100381710038171003817

2

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2119876119894+119904

)

119867

119876119894]

]

= Retr[119877119867119894+119904+1

11986011198751198941198611+ 119877119894+119904+1

119867

11986021198751198941198612

+ 119877119894+119904+1

119861119867

3119875119894119860119867

3+ 119877119894+119904+1

1198614

119867

1198751198941198604

119867

+

1003817100381710038171003817119877119894+119904+11003817100381710038171003817

2

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2119875119867

119894+119904119875119894+ 119877119867

119894+119904+111986211198761198941198631

+ 119877119894+119904+1

119867

11986221198761198941198632+ 119877119894+119904+1

119863119867

3119876119894119862119867

3

+ 119877119894+119904+1

1198634

119867

1198761198941198624

119867

+

1003817100381710038171003817119877119894+119904+11003817100381710038171003817

2

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2119876119867

119894+119904119876119894]

= Retr[119877119867119894+119904+1

11986011198751198941198611+ 119877119894+119904+1

119867

11986021198751198941198612

+ 1198603119875119867

1198941198613119877119867

119894+119904+1+ 1198604119875119879

1198941198614119877119867

119894+119904+1

+

1003817100381710038171003817119877119894+119904+11003817100381710038171003817

2

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2119875119867

119894+119904119875119894+ 119877119867

119894+119904+111986211198761198941198631

+ 119877119894+119904+1

119867

11986221198761198941198632+ 1198623119876119867

1198941198633119877119867

119894+119904+1

+ 1198624119876119879

1198961198634119877119867

119894+119904+1+

1003817100381710038171003817119877119894+119904+11003817100381710038171003817

2

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2119876119867

119894+119904119876119894]

= Retr[119877119867119894+119904+1

(11986011198751198941198611+ 11986211198761198941198631

+ 11986021198751198941198612+ 11986221198761198941198632

+ 1198603119875119867

1198941198613+ 1198623119875119867

1198941198633

+ 1198604119875119879

1198941198614+ 1198624119876119879

1198941198634)

+

1003817100381710038171003817119877119894+119904+11003817100381710038171003817

2

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2(119875119867

119894+119904119875119894+ 119876119867

119894+119904119876119894)]

=

10038171003817100381710038171198751198941003817100381710038171003817

2

+1003817100381710038171003817119876119894

1003817100381710038171003817

2

10038171003817100381710038171198771198941003817100381710038171003817

2Re tr (119877119867

119894+119904+1(119877119894minus 119877119894+1

))

=

10038171003817100381710038171198751198941003817100381710038171003817

2

+1003817100381710038171003817119876119894

1003817100381710038171003817

2

10038171003817100381710038171198771198941003817100381710038171003817

2Re tr (119877119867

119894+119904+1119877119894)

(28)

Also from (19) we have

Re tr (119877119867119894+119904+1

119877119894)

= Re

tr[

[

(119877119894+119904

minus

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2

1003817100381710038171003817119875119894+1199041003817100381710038171003817

2

+1003817100381710038171003817119876119894+119904

1003817100381710038171003817

2

times (1198601119875119894+119904

1198611+ 1198621119876119894+119904

1198631

+ 1198602119875119894+119904

1198612+ 1198622119876119894+119904

1198632

+ 1198603119875119867

119894+1199041198613+ 1198623119876119867

119894+1199041198633

+ 1198604119875119879

119894+1199041198614+ 1198624119876119879

119894+1199041198634))

119867

119877119894]

]

= Retr (119877119867119894+119904

119877119894) minus

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2

1003817100381710038171003817119875119894+1199041003817100381710038171003817

2

+1003817100381710038171003817119876119894+119904

1003817100381710038171003817

2

times tr (119875119867119894+119904

119860119867

1119877119894119861119867

1+ 119876119867

119894+119904119862119867

1119877119894119863119867

1

+ 119875119894+119904

119867

119860119867

2119877119894119861119867

2+ 119876119894+119904

119867

119862119867

2119877119894119863119867

2

+ 119875119894+119904

119860119867

3119877119894119861119867

3+ 119876119894+119904

119862119867

3119877119894119863119867

3

+ 119875119894+119904

119860119867

4119877119894119861119867

4+ 119876119894+119904

119862119867

4119877119894119863119867

4)

Re tr (119877119867119894+119904+1

119877119894)

= minus

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2

1003817100381710038171003817119875119894+1199041003817100381710038171003817

2

+1003817100381710038171003817119876119894+119904

1003817100381710038171003817

2

times Retr(119875119867119894+119904

119860119867

1119877119894119861119867

1+ 119876119867

119894+119904119862119867

1119877119894119863119867

1

+ 119875119894+119904

119867

119860119867

2119877119894119861119867

2+ 119876119894+119904

119867

119862119867

2119877119894119863119867

2

+ 1198613119877119867

1198941198603119875119867

119894+119904+ 1198633119877119867

1198941198623119876119867

119894+119904

+ 1198614119877119879

1198941198604119875119867

119894+119904+ 1198634119877119879

1198941198624119876119867

119894+119904)

Journal of Discrete Mathematics 11

= minus

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2

1003817100381710038171003817119875119894+1199041003817100381710038171003817

2

+1003817100381710038171003817119876119894+119904

1003817100381710038171003817

2

times Re tr [119875119867119894+119904

(119860119867

1119877119894119861119867

1+ 119860119867

2119877119894119861119867

2

+ 1198613119877119867

1198941198603+ 1198614119877119879

1198941198604)

+ 119876119867

119894+119904(119862119867

1119877119894119863119867

1+ 119862119867

2119877119894119863119867

2

+ 1198633119877119867

1198941198623+ 1198634119877119879

1198941198624)]

= minus

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2

1003817100381710038171003817119875119894+1199041003817100381710038171003817

2

+1003817100381710038171003817119876119894+119904

1003817100381710038171003817

2

times Re tr[119875119867119894+119904

(119875119894minus

10038171003817100381710038171198771198941003817100381710038171003817

2

1003817100381710038171003817119877119894minus11003817100381710038171003817

2119875119894minus1

)

+ 119876119867

119894+119904(119876119894minus

10038171003817100381710038171198771198941003817100381710038171003817

2

1003817100381710038171003817119877119894minus11003817100381710038171003817

2119876119894minus1

)]

=

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2

1003817100381710038171003817119875119894+1199041003817100381710038171003817

2

+1003817100381710038171003817119876119894+119904

1003817100381710038171003817

2

10038171003817100381710038171198771198941003817100381710038171003817

2

1003817100381710038171003817119877119894minus11003817100381710038171003817

2

times Re tr (119875119867119894+119904

119875119894minus1

+ 119876119867

119894+119904119876119894minus1

)

(29)

Repeating (28) and (29) one can easily obtain for certain 120572

and 120573

tr (119875119867119894+119904+1

119875119894+ 119876119867

119894+119904+1119876119894) = 120572 [tr (119875119867

119904+11198751+ 119876119867

119904+11198761)]

tr (119877119867119894+119904+1

119877119894) = 120573 [tr (119877119867

119904+11198771)]

(30)

Combining these two relations with (26) implies that (24)holds for 119897 = 119904 + 1 From Steps 1 and 2 the conclusion holdsby the principle of inductionWith the above two lemmas wehave the following theorem

Theorem 6 (see [32]) If the matrix equation (1) is consistentthen a solution can be obtained within finite iteration steps byusing Algorithm 3 for any initial matrices 119881

11198821

4 Numerical Example

In this section we present numerical example to illustrate theapplication of our proposed methods

Example 7 In this example we illustrate our theoreticalresults of Algorithm 3 for solving the system of matrixequation

11986011198811198611+ 11986211198821198631+ 11986021198811198612+ 11986221198821198632+ 11986031198811198671198613

+ 11986231198821198671198633+ 11986041198811198791198614+ 11986241198821198791198634= 119864

(31)

Because of the influence of the error of calculation theresidual 119877(119896) is usually unequal to zero in this process ofthe iteration We regard the matrix 119877(119896) as a zero matrix if119877(119896) lt 10

minus10Given

1198601= [

2 + 3119894 minus119894 1 + 119894

5 1 + 2119894 minus3]

1198602= [

2 + 3119894 minus119894 1 + 119894

5 1 + 2119894 minus3]

1198603= [

0 2 minus 119894 119894

minus1 + 3119894 2 0] 119860

4=[

0 1 minus 3119894 1 + 119894

0 4 + 119894 minus3119894]

1198621= [

1 + 2119894 3 minus 119894 4

minus119894 2119894 minus3] 119862

2=[

3 + 2119894 0 1 + 119894

0 4119894 1 minus 2119894]

1198623= [

1 minus 3119894 2119894 minus3119894

1 2 + 3119894 4119894] 119862

4=[

1 minus 2119894 0 2

3 minus 119894 1 + 119894 minus1]

1198611= [

[

4 + 119894 minus119894

0 1 minus 119894

4119894 2 + 2119894

]

]

1198612= [

[

0 119894

1 + 119894 0

minus1 minus 119894 3119894

]

]

1198613= [

[

0 1

minus3119894 4 + 119894

5 1 + 2119894

]

]

1198614=[

[

3 + 119894 minus1 minus 119894

0 2 minus 119894

minus1 + 119894 2

]

]

1198631= [

[

0 0

1 minus 3119894 minus119894

2119894 minus3119894

]

]

1198632= [

[

0 119894

1 + 119894 0

minus1 minus 119894 3119894

]

]

1198633= [

[

0 1

minus3119894 4 + 119894

5 1 + 2119894

]

]

1198634= [

[

3119894 minus2 + 119894

0 119894

minus2119894 minus4119894

]

]

119864 = [42 + 55119894 115 + 25119894

minus38 minus 119894 132 + 44119894]

(32)

Taking 1198811= [0 0

0 0

0 0

] and 1198821= [0 0

0 0] we apply Algorithm 3

to compute 119881119896119882119896

And iterating 42 steps we get

119881

=[

[

00126 + 18415119894 00827 + 06381119894 11221 minus 09428119894

minus06903 + 10185119894 18818 + 10203119894 09208 + 04569119894

05344 minus 04909119894 09280 + 07169119894 04872 minus 02734119894

]

]

119882

=[

[

04218 minus 09710119894 01763 + 05183119894 11331 minus 00432119894

06273 minus 01216119894 minus03902 + 04313119894 06240 + 09828119894

minus07011 minus 03418119894 03695 + 17627119894 minus03032 + 07073119894

]

]

(33)

which satisfy the matrix equation

11986011198811198611+ 11986211198821198631+ 11986021198811198612+ 11986221198821198632+ 11986031198811198671198613

+ 11986231198821198671198633+ 11986041198811198791198614+ 11986241198821198791198634= 119864

(34)

12 Journal of Discrete Mathematics

log10 rk

4

2

0

minus2

minus4

minus6

minus8

minus100 5 10 15 20 25 30 35 40 45

k (iteration number)

Figure 1The relation between the number of iterations and residualfor the example

With the corresponding residual100381710038171003817100381711987742

1003817100381710038171003817

=10038171003817100381710038171003817119864 minus 119860

1119881421198611minus 1198621119882421198631minus 1198602119881421198612minus 1198622119882421198632

minus 1198603119881119867

421198613minus 1198623119882119867

421198633minus 1198604119881119879

421198614minus 1198624119882119879

421198634

10038171003817100381710038171003817

= 66115 times 10minus11

(35)

5 Conclusions

The above Figure 1 shows the convergence curve for theresidual function 119877(119896) In this paper an iterative algorithmconstructed to solve a complex matrix equation with conju-gate and transpose of two unknowns of the form 119860

11198811198611+

11986211198821198631+11986021198811198612+11986221198821198632+11986031198811198671198613+11986231198821198671198633+11986041198811198791198614+

11986241198821198791198634

= 119864 is presented We proved that the iterativealgorithms always converge to the solution for any initialmatrices We stated and proved some lemmas and theoremswhere the solutions are obtained The proposed methodis illustrated by numerical example where the obtainednumerical results show that our technique is very neat andefficient

References

[1] F Ding P X Liu and J Ding ldquoIterative solutions of thegeneralized Sylvester matrix equations by using the hierarchicalidentification principlerdquo Applied Mathematics and Computa-tion vol 197 no 1 pp 41ndash50 2008

[2] F Ding and T Chen ldquoGradient based iterative algorithmsfor solving a class of matrix equationsrdquo IEEE Transactions onAutomatic Control vol 50 no 8 pp 1216ndash1221 2005

[3] FDing andTChen ldquoHierarchical gradient-based identificationof multivariable discrete-time systemsrdquo Automatica vol 41 no2 pp 315ndash325 2005

[4] F Ding and T Chen ldquoIterative least-squares solutions of cou-pled Sylvester matrix equationsrdquo Systems and Control Lettersvol 54 no 2 pp 95ndash107 2005

[5] F Ding and T Chen ldquoOn iterative solutions of general coupledmatrix equationsrdquo SIAM Journal on Control and Optimizationvol 44 no 6 pp 2269ndash2284 2006

[6] F Ding and T Chen ldquoHierarchical least squares identificationmethods for multivariable systemsrdquo IEEE Transactions onAutomatic Control vol 50 no 3 pp 397ndash402 2005

[7] F Ding and T Chen ldquoPerformance analysis ofmulti-innovationgradient type identification methodsrdquo Automatica vol 43 no1 pp 1ndash14 2007

[8] AGWuX ZengG RDuan andW JWu ldquoIterative solutionsto the extended Sylvester-conjugate matrix equationsrdquo AppliedMathematics and Computation vol 217 no 1 pp 130ndash142 2010

[9] B Zhou G R Duan and Z Y Li ldquoGradient based iterativealgorithm for solving coupled matrix equationsrdquo Systems andControl Letters vol 58 no 5 pp 327ndash333 2009

[10] J H Bevis F J Hall and R E Hartwing ldquoConsimilarity andthe matrix equation119860119883minus119883119861 = 119862rdquo in Current Trends inMatrixTheory pp 51ndash64 North-Holland New York NY USA 1987

[11] R A Horn and C R Johnson Matrix Analysis CambridgeUniversity Press Cambridge UK 1990

[12] H Liping ldquoConsimilarity of quaternion matrices and complexmatricesrdquo Linear Algebra and Its Applications vol 331 no 1ndash3pp 21ndash30 2001

[13] T Jiang X Cheng and L Chen ldquoAn algebraic relation betweenconsimilarity and similarity of complex matrices and its appli-cationsrdquo Journal of Physics A vol 39 no 29 pp 9215ndash92222006

[14] J H Bevis F J Hall and R E Hartwig ldquoThe matrix equation119860119883 minus 119883119861 = 119862 and its special casesrdquo SIAM Journal on MatrixAnalysis and Applications vol 9 no 3 pp 348ndash359 1988

[15] A G Wu G R Duan and H H Yu ldquoOn solutions of thematrix equations 119883119865 minus 119860119883 = 119862 and 119883119865 minus 119860119883 = 119862rdquo AppliedMathematics and Computation vol 183 no 2 pp 932ndash9412006

[16] A Navarra P L Odell and D M Young ldquoRepresentationof the general common solution to the matrix equations11986011198831198611= 1198621 11986021198831198612= 1198622with applicationsrdquo Computers and

Mathematics with Applications vol 41 no 7-8 pp 929ndash9352001

[17] J W Van der Woude ldquoOn the existence of a common solutionX to the matrix equations 119860

119894119883119861119895= 119862 (i j) E Irdquo Linear Algebra

and Its Applications vol 375 no 1ndash3 pp 135ndash145 2003[18] P Bhimasankaram ldquoCommon solutions to the linear matrix

equations 119860119883 = 119862119883119861 = 119863 and EXF = Grdquo Sankhya Series Avol 38 pp 404ndash409 1976

[19] S Kumar Mitra ldquoThe matrix equations 119860119883 = 119862119883119861 = 119863rdquoLinear Algebra and Its Applications vol 59 pp 171ndash181 1984

[20] S K Mitra ldquoA pair of simultaneous linear matrix equations11986011198831198611

= 1198621 11986021198831198612

= 1198622and a matrix programming

problemrdquo Linear Algebra and Its Applications vol 131 pp 107ndash123 1990

[21] M A Ramadan M A Abdel Naby and A M E BayoumildquoOn the explicit solutions of forms of the Sylvester and theYakubovich matrix equationsrdquo Mathematical and ComputerModelling vol 50 no 9-10 pp 1400ndash1408 2009

[22] K W E Chu ldquoSingular value and generalized singular valuedecompositions and the solution of linear matrix equationsrdquoLinear Algebra and Its Applications vol 88-89 pp 83ndash98 1987

Journal of Discrete Mathematics 13

[23] Y X Yuan ldquoThe optimal solution of linear matrix equationby matrix decompositions Mathrdquo Numerica Sinica vol 24 pp165ndash176 2002

[24] A P Liao and Y Lei ldquoLeast-squares solution with the mini-mum-norm for the matrix equation (119860119883119861 119866119883119867) = (119862119863)rdquoComputers and Mathematics with Applications vol 50 no 3-4pp 539ndash549 2005

[25] Y H Liu ldquoRanks of least squares solutions of the matrix equa-tion 119860119883119861 = 119862rdquo Computers and Mathematics with Applicationsvol 55 no 6 pp 1270ndash1278 2008

[26] X Sheng and G Chen ldquoA finite iterative method for solving apair of linear matrix equations (119860119883119861 119862119883119863) = (119864 119865)rdquo AppliedMathematics and Computation vol 189 no 2 pp 1350ndash13582007

[27] A G Wu W Liu and G R Duan ldquoOn the conjugate productof complex polynomial matricesrdquo Mathematical and ComputerModelling vol 53 no 9-10 pp 2031ndash2043 2011

[28] A GWu G FengW Liu and G R Duan ldquoThe complete solu-tion to the Sylvester-polynomial-conjugate matrix equationsrdquoMathematical and Computer Modelling vol 53 no 9-10 pp2044ndash2056 2011

[29] A G Wu B Li Y Zhang and G R Duan ldquoFinite iterativesolutions to coupled Sylvester-conjugate matrix equationsrdquoApplied Mathematical Modelling vol 35 no 3 pp 1065ndash10802011

[30] AGWu L Lv andGRDuan ldquoIterative algorithms for solvinga class of complex conjugate and transpose matrix equationsrdquoApplied Mathematics and Computation vol 217 no 21 pp8343ndash8353 2011

[31] X ZhangMatrix Analysis and Application TsinghuaUniversityPress Beijing China 2004

[32] A G Wu L Lv and M Z Hou ldquoFinite iterative algorithms forextended Sylvester-conjugate matrix equationsrdquo Mathematicaland Computer Modelling vol 54 pp 2363ndash2384 2011

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Journal of Discrete Mathematics

minus 1198624(119882119896+

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2119876119896)

119879

1198634

= 119864 minus 11986011198811198961198611minus 11986211198821198961198631minus 11986021198811198961198612minus 11986221198821198961198632

minus 1198603119881119867

1198961198613minus 1198623119882119867

1198961198633minus 1198604119881119879

1198961198614

minus 1198624119882119879

1198961198634minus

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

times (11986011198751198961198611+ 11986211198761198961198631+ 11986021198751198961198612

+ 11986221198761198961198632+ 1198603119875119867

1198961198613

+ 1198623119876119867

1198961198633+ 1198604119875119879

1198961198614+ 1198624119876119879

1198961198634)

119877119896+1

= 119877119896minus

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

times (11986011198751198961198611+ 11986211198761198961198631+ 11986021198751198961198612

+ 11986221198761198961198632+ 1198603119875119867

1198961198613

+ 1198623119876119867

1198961198633+ 1198604119875119879

1198961198614+ 1198624119876119879

1198961198634)

(19)

For 119894 = 1 it follows from (19) that

Re tr (11987711986721198771)

= Re

tr[

[

(1198771minus

100381710038171003817100381711987711003817100381710038171003817

2

100381710038171003817100381711987511003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2

times (119860111987511198611+ 119862111987611198631

+ 119860211987511198612+ 119862211987611198632

+ 1198603119875119867

11198613+ 1198623119876119867

11198633

+ 1198604119875119879

11198614+ 1198624119876119879

11198634))

119867

1198771]

]

= Retr (11987711986711198771) minus

100381710038171003817100381711987711003817100381710038171003817

2

100381710038171003817100381711987511003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2

times tr (1198751198671119860119867

11198771119861119867

1+ 119876119867

1119862119867

11198771119863119867

1

+ 1198751

119867

119860119867

21198771119861119867

2+ 1198761

119867

119862119867

21198771119863119867

2

+ 1198751119860119867

31198771119861119867

3+ 1198761119862119867

31198771119863119867

3

+ 1198751119860119867

41198771119861119867

4+ 1198761119862119867

41198771119863119867

4)

=10038171003817100381710038171198771

1003817100381710038171003817

2

minus

100381710038171003817100381711987711003817100381710038171003817

2

100381710038171003817100381711987511003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2

times Re tr [1198751198671119860119867

11198771119861119867

1+ 119876119867

1119862119867

11198771119863119867

1

+ 1198751

119867

119860119867

21198771119861119867

2+ 1198761

119867

119862119867

21198771119863119867

2

+ 1198613119877119867

11198603119875119867

1+ 1198633119877119867

11198623119876119867

1

+ 1198614119877119879

11198604119875119867

1+ 1198634119877119879

11198624119876119867

1]

=10038171003817100381710038171198771

1003817100381710038171003817

2

minus

100381710038171003817100381711987711003817100381710038171003817

2

100381710038171003817100381711987511003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2

times Re tr [1198751198671

(119860119867

11198771119861119867

1+ 119860119867

21198771119861119867

2

+ 1198613119877119867

11198603+ 1198614119877119879

11198604)

+ 119876119867

1(119862119867

11198771119863119867

1+ 119862119867

21198771119863119867

2

+ 1198633119877119867

11198623+ 1198634119877119879

11198624)]

=10038171003817100381710038171198771

1003817100381710038171003817

2

minus

100381710038171003817100381711987711003817100381710038171003817

2

100381710038171003817100381711987511003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2Re tr (119875119867

11198751+ 119876119867

11198761)

Re tr (11987711986721198771)

=10038171003817100381710038171198771

1003817100381710038171003817

2

minus

100381710038171003817100381711987711003817100381710038171003817

2

100381710038171003817100381711987511003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2(10038171003817100381710038171198751

1003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2

) = 0

(20)

This implies that (17) is satisfied for 119894 = 1From Algorithm 3 we have

Re tr (11987511986721198751+ 119876119867

21198761)

= Re

tr[

[

(119860119867

11198772119861119867

1+ 119860119867

21198772119861119867

2+ 1198613119877119867

21198603

+ 1198614119877119879

21198604+

100381710038171003817100381711987721003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

21198751)

119867

1198751

+ (119862119867

11198772119863119867

1+ 119862119867

21198772119863119867

2+ 1198633119877119867

21198623

+1198634119877119879

21198624+

100381710038171003817100381711987721003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

21198761)

119867

1198761]

]

Journal of Discrete Mathematics 7

= Retr[1198771198672119860111987511198611+ 1198772

119867

119860211987511198612+ 1198772119861119867

31198751119860119867

3

+ 11987721198614

119867

11987511198604

119867

+

100381710038171003817100381711987721003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

2119875119867

11198751+ 119877119867

2119862111987611198631

+ 1198772

119867

119862211987611198632+ 1198772119863119867

31198761119862119867

3

+ 11987721198634

119867

11987611198624

119867

+

100381710038171003817100381711987721003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

2119876119867

11198761]

= Retr[1198771198672119860111987511198611+ 1198772

119867

119860211987511198612+ 1198603119875119867

11198613119877119867

2

+ 1198604119875119879

11198614119877119867

2+ 119877119867

2119862111987611198631

+ 1198772

119867

119862211987611198632+ 1198623119876119867

11198633119877119867

2

+ 1198624119876119879

11198634119877119867

2+

100381710038171003817100381711987721003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

2(119875119867

11198751+ 119876119867

11198761)]

= Retr[1198771198672119860111987511198611+ 119877119867

2119860211987511198612+ 119877119867

21198603119875119867

11198613

+ 119877119867

21198604119875119879

11198614+ 119877119867

2119862111987611198631

+ 119877119867

2119862211987611198632+ 119877119867

21198623119876119867

11198633

+ 119877119867

21198624119876119879

11198634+

100381710038171003817100381711987721003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

2(119875119867

11198751+ 119876119867

11198761)]

= Retr[1198771198672(119860111987511198611+ 119862111987611198631+ 119860211987511198612

+ 119862211987611198632+ 1198603119875119867

11198613+ 1198623119876119867

11198633

+1198604119875119879

11198614+ 1198624119876119867

11198634) ]

+

100381710038171003817100381711987721003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

2(10038171003817100381710038171198751

1003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2

)

=

100381710038171003817100381711987511003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

2Re tr [119877119867

2(1198771minus 1198772)]

+

100381710038171003817100381711987721003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

2(10038171003817100381710038171198751

1003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2

)

= minus

100381710038171003817100381711987511003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

2(10038171003817100381710038171198772

1003817100381710038171003817

2

)

+

100381710038171003817100381711987721003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

2(10038171003817100381710038171198751

1003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2

) = 0

(21)This implies that (18) is satisfied for 119894 = 1

Assume that (17) and (18) hold for 119894 = 119896 minus 1 fromAlgorithm 3 we have

Re tr (119877119867119896+1

119877119896)

= Re

tr[

[

(119877119896minus

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

times (11986011198751198961198611+ 11986211198761198961198631+ 11986021198751198961198612

+ 11986221198761198961198632+ 1198603119875119867

1198961198613+ 1198623119876119867

1198961198633

+ 1198604119875119879

1198961198614+ 1198624119876119879

1198961198634))

119867

119877119896]

]

Re tr (119877119867119896+1

119877119896)

= Retr (119877119867119896119877119896) minus

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

times tr (119875119867119896119860119867

1119877119896119861119867

1+ 119876119867

119896119862119867

1119877119896119863119867

1

+ 119875119896

119867

119860119867

2119877119896119861119867

2+ 119876119896

119867

119862119867

2119877119896119863119867

2

+ 119875119896119860119867

3119877119896119861119867

3+ 119876119896119862119867

3119877119896119863119867

3

+ 119875119896119860119867

4119877119896119861119867

4+ 119876119896119862119867

4119877119896119863119867

4)

=1003817100381710038171003817119877119896

1003817100381710038171003817

2

minus

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

times Re tr (119875119867119896119860119867

1119877119896119861119867

1+ 119876119867

119896119862119867

1119877119896119863119867

1

+ 119875119896

119867

119860119867

2119877119896119861119867

2+ 119876119896

119867

119862119867

2119877119896119863119867

2

+ 1198613119877119867

1198961198603119875119867

119896+ 1198633119877119867

1198961198623119876119867

119896

+ 1198614119877119879

1198961198604119875119867

119896+ 1198634119877119879

1198961198624119876119867

119896)

=10038171003817100381710038171198771

1003817100381710038171003817

2

minus

100381710038171003817100381711987711003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

times Re tr [119875119867119896

(119860119867

1119877119896119861119867

1+ 119860119867

2119877119896119861119867

2

+ 1198613119877119867

1198961198603+ 1198614119877119879

1198961198604)

+ 119876119867

119896(119862119867

1119877119896119863119867

1+ 119862119867

2119877119896119863119867

2

+ 1198633119877119867

1198961198623+ 1198634119877119879

1198961198624)]

8 Journal of Discrete Mathematics

=1003817100381710038171003817119877119896

1003817100381710038171003817

2

minus

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

times Retr[119875119867119896

(119875119896minus

10038171003817100381710038171198771198961003817100381710038171003817

2

1003817100381710038171003817119877119896minus11003817100381710038171003817

2119875119896minus1

)

+ 119876119867

119896(119876119896minus

10038171003817100381710038171198771198961003817100381710038171003817

2

1003817100381710038171003817119877119896minus11003817100381710038171003817

2119876119896minus1

)]

=1003817100381710038171003817119877119896

1003817100381710038171003817

2

minus

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2(1003817100381710038171003817119875119896

1003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

) = 0

(22)

Thus (17) holds for 119894 = 119896Also from Algorithm 3 we have

Re tr (119875119867119896+1

119875119896+ 119876119867

119896+1119876119896)

= Re

tr [

[

(119860119867

1119877119896+1

119861119867

1+ 119860119867

2119877119896+1

119861119867

2

+ 1198613119877119867

119896+11198603+ 1198614119877119879

119896+11198604

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2119875119896)

119867

119875119896

+ (119862119867

1119877119896+1

119863119867

1+ 119862119867

2119877119896+1

119863119867

2

+ 1198633119877119867

119896+11198623+ 1198634119877119879

119896+11198624

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2119876119896)

119867

119876119896]

]

= Retr[119877119867119896+1

11986011198751198961198611+ 119877119896+1

119867

11986021198751198961198612

+ 119877119896+1

119861119867

3119875119896119860119867

3+ 119877119896+1

1198614

119867

1198751198961198604

119867

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2119875119867

119896119875119896+ 119877119867

119896+111986211198761198961198631

+ 119877119896+1

119867

11986221198761198961198632+ 119877119896+1

119863119867

3119876119896119862119867

3

+ 119877119896+1

1198634

119867

1198761198961198624

119867

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2119876119867

119896119876119896]

= Retr[119877119867119896+1

11986011198751198961198611+ 119877119896+1

119867

11986021198751198961198612

+ 1198603119875119867

1198961198613119877119867

119896+1+ 1198604119875119879

1198961198614119877119867

119896+1

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2119875119867

119896119875119896+ 119877119867

119896+111986211198761198961198631

+ 119877119896+1

119867

11986221198761198961198632+ 1198623119876119867

1198961198633119877119867

119896+1

+ 1198624119876119879

1198961198634119877119867

119896+1+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2119876119867

119896119876119896]

= Retr[119877119867119896+1

(11986011198751198961198611+ 11986211198761198961198631

+ 11986021198751198961198612+ 11986221198761198961198632+ 1198603119875119867

1198961198613

+1198623119876119867

1198961198633+ 1198604119875119879

1198961198614+ 1198624119876119879

1198961198634)

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2(119875119867

119896119875119896+ 119876119867

119896119876119896)]

=

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2Re tr (119877119867

119896+1(119877119896minus 119877119896+1

))

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2(1003817100381710038171003817119875119896

1003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

)

= minus

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2(1003817100381710038171003817119877119896+1

1003817100381710038171003817

2

)

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2(1003817100381710038171003817119875119896

1003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

) = 0

(23)

This implies that (17) and (18) hold for 119894 = 119896Then relations (17) and (18) holds bymathematical induc-

tionStep 2 We want to show that

Re (tr (119877119867119894+119897119877119894)) = 0

Re (tr (119875119867119894+119897119875119894+ 119876119867

119894+119897119876119894)) = 0

(24)

holds for 119897 ge 1 We will prove this conclusion by inductionThe case of 119897 = 1 has been proven in Step 1 Now we assumethat (24) holds for 119897 le 119904 119904 ge 1 The aim is to show that

Re (tr (119877119867119894+119904+1

119877119894)) = 0

Re (tr (119875119867119894+119904+1

119875119894+ 119876119867

119894+119904+1119876119894)) = 0

(25)

First we prove the following

Re (tr (119877119867119904+1

1198770)) = 0

Re (tr (119875119867119904+1

1198750+ 119876119867

119904+11198760)) = 0

(26)

Journal of Discrete Mathematics 9

By using Algorithm 3 from (19) and induction we have

Re tr (119877119867119904+1

1198770)

= Re

tr[

[

(119877119904minus

10038171003817100381710038171198771199041003817100381710038171003817

2

10038171003817100381710038171198751199041003817100381710038171003817

2

+1003817100381710038171003817119876119904

1003817100381710038171003817

2

times (11986011198751199041198611+ 11986211198761199041198631+ 11986021198751199041198612

+ 11986221198761199041198632+ 1198603119875119867

1199041198613+ 1198623119876119867

1199041198633

+ 1198604119875119879

1199041198614+ 1198624119876119879

1199041198634))

119867

1198770]

]

= Re tr (1198771198671199041198770) minus

10038171003817100381710038171198771199041003817100381710038171003817

2

10038171003817100381710038171198751199041003817100381710038171003817

2

+1003817100381710038171003817119876119904

1003817100381710038171003817

2

times tr (119875119867119904119860119867

11198770119861119867

1+ 119876119867

119904119862119867

11198770119863119867

1

+ 119875119904

119867

119860119867

21198770119861119867

2+ 119876119904

119867

119862119867

21198770119863119867

2

+ 119875119904119860119867

31198770119861119867

3+ 119876119904119862119867

31198770119863119867

3

+ 119875119904119860119867

41198770119861119867

4+ 119876119904119862119867

41198770119863119867

4)

Re tr (119877119867119904+1

1198770)

= Re tr (1198771198671199041198770) minus

10038171003817100381710038171198771199041003817100381710038171003817

2

10038171003817100381710038171198751199041003817100381710038171003817

2

+1003817100381710038171003817119876119904

1003817100381710038171003817

2

times Re tr (119875119867119904119860119867

11198770119861119867

1+ 119876119867

119904119862119867

11198770119863119867

1

+ 119875119904

119867

119860119867

21198770119861119867

2+ 119876119904

119867

119862119867

21198770119863119867

2

+ 1198613119877119867

01198603119875119867

119904+ 1198633119877119867

01198623119876119867

119904

+ 1198614119877119879

01198604119875119867

119904+ 1198634119877119879

01198624119876119867

119904)

= minus

10038171003817100381710038171198771199041003817100381710038171003817

2

10038171003817100381710038171198751199041003817100381710038171003817

2

+1003817100381710038171003817119876119904

1003817100381710038171003817

2

times Re tr (119875119867119904

(119860119867

11198770119861119867

1+ 119860119867

21198770119861119867

2

+ 1198613119877119867

01198603+ 1198614119877119879

01198604)

+ 119876119867

119904(119862119867

11198770119863119867

1+ 119862119867

21198770119863119867

2

+1198633119877119867

01198623+ 1198634119877119879

01198624))

= minus

10038171003817100381710038171198771199041003817100381710038171003817

2

10038171003817100381710038171198751199041003817100381710038171003817

2

+1003817100381710038171003817119876119904

1003817100381710038171003817

2Re tr (119875119867

1199041198750+ 119876119867

1199041198760) = 0

Re tr (119875119867119904+1

1198750+ 119876119867

119904+11198760)

= Retr[(119860119867

1119877119904+1

119861119867

1+ 119860119867

2119877119904+1

119861119867

2

+ 1198613119877119867

119904+11198603+ 1198614119877119879

119904+11198604

+

1003817100381710038171003817119877119904+11003817100381710038171003817

2

10038171003817100381710038171198771199041003817100381710038171003817

2119875119904)

119867

1198750

+ (119862119867

1119877119904+1

119863119867

1+ 119862119867

2119877119904+1

119863119867

2

+ 1198633119877119867

119904+11198623+ 1198634119877119879

119896+11198624

+

1003817100381710038171003817119877119904+11003817100381710038171003817

2

10038171003817100381710038171198771199041003817100381710038171003817

2119876119904)

119867

1198760]

]

= Retr[119877119867119904+1

119860111987501198611+ 119877119904+1

119867

119860211987501198612

+ 119877119904+1

119861119867

31198750119860119867

3+ 119877119904+1

1198614

119867

11987501198604

119867

+

1003817100381710038171003817119877119904+11003817100381710038171003817

2

10038171003817100381710038171198771199041003817100381710038171003817

2119875119867

1199041198750+ 119877119867

119904+1119862111987601198631

+ 119877119904+1

119867

119862211987601198632+ 119877119904+1

119863119867

31198760119862119867

3

+ 119877119904+1

1198634

119867

11987601198624

119867

+

1003817100381710038171003817119877119904+11003817100381710038171003817

2

10038171003817100381710038171198771199041003817100381710038171003817

2119876119867

1199041198760]

= Retr[119877119867119904+1

119860111987501198611+ 119877119904+1

119867

119860211987501198612

+ 1198603119875119867

01198613119877119867

119904+1+ 1198604119875119879

01198614119877119867

119904+1

+

1003817100381710038171003817119877119904+11003817100381710038171003817

2

10038171003817100381710038171198771199041003817100381710038171003817

2119875119867

1199041198750+ 119877119867

119904+1119862111987601198631

+ 119877119904+1

119867

119862211987601198632+ 1198623119876119867

01198633119877119867

119904+1

+ 1198624119876119879

01198634119877119867

119904+1+

1003817100381710038171003817119877119904+11003817100381710038171003817

2

10038171003817100381710038171198771199041003817100381710038171003817

2119876119867

1199041198760]

= Retr[119877119867119904+1

(119860111987501198611+ 119862111987601198631+ 119860211987501198612

+ 119862211987601198632+ 1198603119875119867

01198613

+ 1198623119876119867

01198633+ 1198604119875119879

01198614

+ 1198624119876119879

01198634)

+

1003817100381710038171003817119877119904+11003817100381710038171003817

2

10038171003817100381710038171198771199041003817100381710038171003817

2(119875119867

1199041198750+ 119876119867

1199041198760)]

=

100381710038171003817100381711987501003817100381710038171003817

2

+10038171003817100381710038171198760

1003817100381710038171003817

2

100381710038171003817100381711987701003817100381710038171003817

2Re tr (119877119867

119904+1(1198770minus 1198771))= 0

(27)

Then (26) is holds

10 Journal of Discrete Mathematics

From Algorithm 3 we have

Re tr (119875119867119894+119904+1

119875119894+ 119876119867

119894+119904+1119876119894)

= Retr[(119860119867

1119877119894+119904+1

119861119867

1+ 119860119867

2119877119894+119904+1

119861119867

2

+ 1198613119877119867

119894+119904+11198603+ 1198614119877119879

119894+119904+11198604

+

1003817100381710038171003817119877119894+119904+11003817100381710038171003817

2

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2119875119894+119904

)

119867

119875119894

+ (119862119867

1119877119894+119904+1

119863119867

1+ 119862119867

2119877119894+119904+1

119863119867

2

+ 1198633119877119867

119894+119904+11198623+ 1198634119877119879

119894+119904+11198624

+

1003817100381710038171003817119877119894+119904+11003817100381710038171003817

2

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2119876119894+119904

)

119867

119876119894]

]

= Retr[119877119867119894+119904+1

11986011198751198941198611+ 119877119894+119904+1

119867

11986021198751198941198612

+ 119877119894+119904+1

119861119867

3119875119894119860119867

3+ 119877119894+119904+1

1198614

119867

1198751198941198604

119867

+

1003817100381710038171003817119877119894+119904+11003817100381710038171003817

2

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2119875119867

119894+119904119875119894+ 119877119867

119894+119904+111986211198761198941198631

+ 119877119894+119904+1

119867

11986221198761198941198632+ 119877119894+119904+1

119863119867

3119876119894119862119867

3

+ 119877119894+119904+1

1198634

119867

1198761198941198624

119867

+

1003817100381710038171003817119877119894+119904+11003817100381710038171003817

2

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2119876119867

119894+119904119876119894]

= Retr[119877119867119894+119904+1

11986011198751198941198611+ 119877119894+119904+1

119867

11986021198751198941198612

+ 1198603119875119867

1198941198613119877119867

119894+119904+1+ 1198604119875119879

1198941198614119877119867

119894+119904+1

+

1003817100381710038171003817119877119894+119904+11003817100381710038171003817

2

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2119875119867

119894+119904119875119894+ 119877119867

119894+119904+111986211198761198941198631

+ 119877119894+119904+1

119867

11986221198761198941198632+ 1198623119876119867

1198941198633119877119867

119894+119904+1

+ 1198624119876119879

1198961198634119877119867

119894+119904+1+

1003817100381710038171003817119877119894+119904+11003817100381710038171003817

2

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2119876119867

119894+119904119876119894]

= Retr[119877119867119894+119904+1

(11986011198751198941198611+ 11986211198761198941198631

+ 11986021198751198941198612+ 11986221198761198941198632

+ 1198603119875119867

1198941198613+ 1198623119875119867

1198941198633

+ 1198604119875119879

1198941198614+ 1198624119876119879

1198941198634)

+

1003817100381710038171003817119877119894+119904+11003817100381710038171003817

2

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2(119875119867

119894+119904119875119894+ 119876119867

119894+119904119876119894)]

=

10038171003817100381710038171198751198941003817100381710038171003817

2

+1003817100381710038171003817119876119894

1003817100381710038171003817

2

10038171003817100381710038171198771198941003817100381710038171003817

2Re tr (119877119867

119894+119904+1(119877119894minus 119877119894+1

))

=

10038171003817100381710038171198751198941003817100381710038171003817

2

+1003817100381710038171003817119876119894

1003817100381710038171003817

2

10038171003817100381710038171198771198941003817100381710038171003817

2Re tr (119877119867

119894+119904+1119877119894)

(28)

Also from (19) we have

Re tr (119877119867119894+119904+1

119877119894)

= Re

tr[

[

(119877119894+119904

minus

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2

1003817100381710038171003817119875119894+1199041003817100381710038171003817

2

+1003817100381710038171003817119876119894+119904

1003817100381710038171003817

2

times (1198601119875119894+119904

1198611+ 1198621119876119894+119904

1198631

+ 1198602119875119894+119904

1198612+ 1198622119876119894+119904

1198632

+ 1198603119875119867

119894+1199041198613+ 1198623119876119867

119894+1199041198633

+ 1198604119875119879

119894+1199041198614+ 1198624119876119879

119894+1199041198634))

119867

119877119894]

]

= Retr (119877119867119894+119904

119877119894) minus

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2

1003817100381710038171003817119875119894+1199041003817100381710038171003817

2

+1003817100381710038171003817119876119894+119904

1003817100381710038171003817

2

times tr (119875119867119894+119904

119860119867

1119877119894119861119867

1+ 119876119867

119894+119904119862119867

1119877119894119863119867

1

+ 119875119894+119904

119867

119860119867

2119877119894119861119867

2+ 119876119894+119904

119867

119862119867

2119877119894119863119867

2

+ 119875119894+119904

119860119867

3119877119894119861119867

3+ 119876119894+119904

119862119867

3119877119894119863119867

3

+ 119875119894+119904

119860119867

4119877119894119861119867

4+ 119876119894+119904

119862119867

4119877119894119863119867

4)

Re tr (119877119867119894+119904+1

119877119894)

= minus

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2

1003817100381710038171003817119875119894+1199041003817100381710038171003817

2

+1003817100381710038171003817119876119894+119904

1003817100381710038171003817

2

times Retr(119875119867119894+119904

119860119867

1119877119894119861119867

1+ 119876119867

119894+119904119862119867

1119877119894119863119867

1

+ 119875119894+119904

119867

119860119867

2119877119894119861119867

2+ 119876119894+119904

119867

119862119867

2119877119894119863119867

2

+ 1198613119877119867

1198941198603119875119867

119894+119904+ 1198633119877119867

1198941198623119876119867

119894+119904

+ 1198614119877119879

1198941198604119875119867

119894+119904+ 1198634119877119879

1198941198624119876119867

119894+119904)

Journal of Discrete Mathematics 11

= minus

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2

1003817100381710038171003817119875119894+1199041003817100381710038171003817

2

+1003817100381710038171003817119876119894+119904

1003817100381710038171003817

2

times Re tr [119875119867119894+119904

(119860119867

1119877119894119861119867

1+ 119860119867

2119877119894119861119867

2

+ 1198613119877119867

1198941198603+ 1198614119877119879

1198941198604)

+ 119876119867

119894+119904(119862119867

1119877119894119863119867

1+ 119862119867

2119877119894119863119867

2

+ 1198633119877119867

1198941198623+ 1198634119877119879

1198941198624)]

= minus

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2

1003817100381710038171003817119875119894+1199041003817100381710038171003817

2

+1003817100381710038171003817119876119894+119904

1003817100381710038171003817

2

times Re tr[119875119867119894+119904

(119875119894minus

10038171003817100381710038171198771198941003817100381710038171003817

2

1003817100381710038171003817119877119894minus11003817100381710038171003817

2119875119894minus1

)

+ 119876119867

119894+119904(119876119894minus

10038171003817100381710038171198771198941003817100381710038171003817

2

1003817100381710038171003817119877119894minus11003817100381710038171003817

2119876119894minus1

)]

=

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2

1003817100381710038171003817119875119894+1199041003817100381710038171003817

2

+1003817100381710038171003817119876119894+119904

1003817100381710038171003817

2

10038171003817100381710038171198771198941003817100381710038171003817

2

1003817100381710038171003817119877119894minus11003817100381710038171003817

2

times Re tr (119875119867119894+119904

119875119894minus1

+ 119876119867

119894+119904119876119894minus1

)

(29)

Repeating (28) and (29) one can easily obtain for certain 120572

and 120573

tr (119875119867119894+119904+1

119875119894+ 119876119867

119894+119904+1119876119894) = 120572 [tr (119875119867

119904+11198751+ 119876119867

119904+11198761)]

tr (119877119867119894+119904+1

119877119894) = 120573 [tr (119877119867

119904+11198771)]

(30)

Combining these two relations with (26) implies that (24)holds for 119897 = 119904 + 1 From Steps 1 and 2 the conclusion holdsby the principle of inductionWith the above two lemmas wehave the following theorem

Theorem 6 (see [32]) If the matrix equation (1) is consistentthen a solution can be obtained within finite iteration steps byusing Algorithm 3 for any initial matrices 119881

11198821

4 Numerical Example

In this section we present numerical example to illustrate theapplication of our proposed methods

Example 7 In this example we illustrate our theoreticalresults of Algorithm 3 for solving the system of matrixequation

11986011198811198611+ 11986211198821198631+ 11986021198811198612+ 11986221198821198632+ 11986031198811198671198613

+ 11986231198821198671198633+ 11986041198811198791198614+ 11986241198821198791198634= 119864

(31)

Because of the influence of the error of calculation theresidual 119877(119896) is usually unequal to zero in this process ofthe iteration We regard the matrix 119877(119896) as a zero matrix if119877(119896) lt 10

minus10Given

1198601= [

2 + 3119894 minus119894 1 + 119894

5 1 + 2119894 minus3]

1198602= [

2 + 3119894 minus119894 1 + 119894

5 1 + 2119894 minus3]

1198603= [

0 2 minus 119894 119894

minus1 + 3119894 2 0] 119860

4=[

0 1 minus 3119894 1 + 119894

0 4 + 119894 minus3119894]

1198621= [

1 + 2119894 3 minus 119894 4

minus119894 2119894 minus3] 119862

2=[

3 + 2119894 0 1 + 119894

0 4119894 1 minus 2119894]

1198623= [

1 minus 3119894 2119894 minus3119894

1 2 + 3119894 4119894] 119862

4=[

1 minus 2119894 0 2

3 minus 119894 1 + 119894 minus1]

1198611= [

[

4 + 119894 minus119894

0 1 minus 119894

4119894 2 + 2119894

]

]

1198612= [

[

0 119894

1 + 119894 0

minus1 minus 119894 3119894

]

]

1198613= [

[

0 1

minus3119894 4 + 119894

5 1 + 2119894

]

]

1198614=[

[

3 + 119894 minus1 minus 119894

0 2 minus 119894

minus1 + 119894 2

]

]

1198631= [

[

0 0

1 minus 3119894 minus119894

2119894 minus3119894

]

]

1198632= [

[

0 119894

1 + 119894 0

minus1 minus 119894 3119894

]

]

1198633= [

[

0 1

minus3119894 4 + 119894

5 1 + 2119894

]

]

1198634= [

[

3119894 minus2 + 119894

0 119894

minus2119894 minus4119894

]

]

119864 = [42 + 55119894 115 + 25119894

minus38 minus 119894 132 + 44119894]

(32)

Taking 1198811= [0 0

0 0

0 0

] and 1198821= [0 0

0 0] we apply Algorithm 3

to compute 119881119896119882119896

And iterating 42 steps we get

119881

=[

[

00126 + 18415119894 00827 + 06381119894 11221 minus 09428119894

minus06903 + 10185119894 18818 + 10203119894 09208 + 04569119894

05344 minus 04909119894 09280 + 07169119894 04872 minus 02734119894

]

]

119882

=[

[

04218 minus 09710119894 01763 + 05183119894 11331 minus 00432119894

06273 minus 01216119894 minus03902 + 04313119894 06240 + 09828119894

minus07011 minus 03418119894 03695 + 17627119894 minus03032 + 07073119894

]

]

(33)

which satisfy the matrix equation

11986011198811198611+ 11986211198821198631+ 11986021198811198612+ 11986221198821198632+ 11986031198811198671198613

+ 11986231198821198671198633+ 11986041198811198791198614+ 11986241198821198791198634= 119864

(34)

12 Journal of Discrete Mathematics

log10 rk

4

2

0

minus2

minus4

minus6

minus8

minus100 5 10 15 20 25 30 35 40 45

k (iteration number)

Figure 1The relation between the number of iterations and residualfor the example

With the corresponding residual100381710038171003817100381711987742

1003817100381710038171003817

=10038171003817100381710038171003817119864 minus 119860

1119881421198611minus 1198621119882421198631minus 1198602119881421198612minus 1198622119882421198632

minus 1198603119881119867

421198613minus 1198623119882119867

421198633minus 1198604119881119879

421198614minus 1198624119882119879

421198634

10038171003817100381710038171003817

= 66115 times 10minus11

(35)

5 Conclusions

The above Figure 1 shows the convergence curve for theresidual function 119877(119896) In this paper an iterative algorithmconstructed to solve a complex matrix equation with conju-gate and transpose of two unknowns of the form 119860

11198811198611+

11986211198821198631+11986021198811198612+11986221198821198632+11986031198811198671198613+11986231198821198671198633+11986041198811198791198614+

11986241198821198791198634

= 119864 is presented We proved that the iterativealgorithms always converge to the solution for any initialmatrices We stated and proved some lemmas and theoremswhere the solutions are obtained The proposed methodis illustrated by numerical example where the obtainednumerical results show that our technique is very neat andefficient

References

[1] F Ding P X Liu and J Ding ldquoIterative solutions of thegeneralized Sylvester matrix equations by using the hierarchicalidentification principlerdquo Applied Mathematics and Computa-tion vol 197 no 1 pp 41ndash50 2008

[2] F Ding and T Chen ldquoGradient based iterative algorithmsfor solving a class of matrix equationsrdquo IEEE Transactions onAutomatic Control vol 50 no 8 pp 1216ndash1221 2005

[3] FDing andTChen ldquoHierarchical gradient-based identificationof multivariable discrete-time systemsrdquo Automatica vol 41 no2 pp 315ndash325 2005

[4] F Ding and T Chen ldquoIterative least-squares solutions of cou-pled Sylvester matrix equationsrdquo Systems and Control Lettersvol 54 no 2 pp 95ndash107 2005

[5] F Ding and T Chen ldquoOn iterative solutions of general coupledmatrix equationsrdquo SIAM Journal on Control and Optimizationvol 44 no 6 pp 2269ndash2284 2006

[6] F Ding and T Chen ldquoHierarchical least squares identificationmethods for multivariable systemsrdquo IEEE Transactions onAutomatic Control vol 50 no 3 pp 397ndash402 2005

[7] F Ding and T Chen ldquoPerformance analysis ofmulti-innovationgradient type identification methodsrdquo Automatica vol 43 no1 pp 1ndash14 2007

[8] AGWuX ZengG RDuan andW JWu ldquoIterative solutionsto the extended Sylvester-conjugate matrix equationsrdquo AppliedMathematics and Computation vol 217 no 1 pp 130ndash142 2010

[9] B Zhou G R Duan and Z Y Li ldquoGradient based iterativealgorithm for solving coupled matrix equationsrdquo Systems andControl Letters vol 58 no 5 pp 327ndash333 2009

[10] J H Bevis F J Hall and R E Hartwing ldquoConsimilarity andthe matrix equation119860119883minus119883119861 = 119862rdquo in Current Trends inMatrixTheory pp 51ndash64 North-Holland New York NY USA 1987

[11] R A Horn and C R Johnson Matrix Analysis CambridgeUniversity Press Cambridge UK 1990

[12] H Liping ldquoConsimilarity of quaternion matrices and complexmatricesrdquo Linear Algebra and Its Applications vol 331 no 1ndash3pp 21ndash30 2001

[13] T Jiang X Cheng and L Chen ldquoAn algebraic relation betweenconsimilarity and similarity of complex matrices and its appli-cationsrdquo Journal of Physics A vol 39 no 29 pp 9215ndash92222006

[14] J H Bevis F J Hall and R E Hartwig ldquoThe matrix equation119860119883 minus 119883119861 = 119862 and its special casesrdquo SIAM Journal on MatrixAnalysis and Applications vol 9 no 3 pp 348ndash359 1988

[15] A G Wu G R Duan and H H Yu ldquoOn solutions of thematrix equations 119883119865 minus 119860119883 = 119862 and 119883119865 minus 119860119883 = 119862rdquo AppliedMathematics and Computation vol 183 no 2 pp 932ndash9412006

[16] A Navarra P L Odell and D M Young ldquoRepresentationof the general common solution to the matrix equations11986011198831198611= 1198621 11986021198831198612= 1198622with applicationsrdquo Computers and

Mathematics with Applications vol 41 no 7-8 pp 929ndash9352001

[17] J W Van der Woude ldquoOn the existence of a common solutionX to the matrix equations 119860

119894119883119861119895= 119862 (i j) E Irdquo Linear Algebra

and Its Applications vol 375 no 1ndash3 pp 135ndash145 2003[18] P Bhimasankaram ldquoCommon solutions to the linear matrix

equations 119860119883 = 119862119883119861 = 119863 and EXF = Grdquo Sankhya Series Avol 38 pp 404ndash409 1976

[19] S Kumar Mitra ldquoThe matrix equations 119860119883 = 119862119883119861 = 119863rdquoLinear Algebra and Its Applications vol 59 pp 171ndash181 1984

[20] S K Mitra ldquoA pair of simultaneous linear matrix equations11986011198831198611

= 1198621 11986021198831198612

= 1198622and a matrix programming

problemrdquo Linear Algebra and Its Applications vol 131 pp 107ndash123 1990

[21] M A Ramadan M A Abdel Naby and A M E BayoumildquoOn the explicit solutions of forms of the Sylvester and theYakubovich matrix equationsrdquo Mathematical and ComputerModelling vol 50 no 9-10 pp 1400ndash1408 2009

[22] K W E Chu ldquoSingular value and generalized singular valuedecompositions and the solution of linear matrix equationsrdquoLinear Algebra and Its Applications vol 88-89 pp 83ndash98 1987

Journal of Discrete Mathematics 13

[23] Y X Yuan ldquoThe optimal solution of linear matrix equationby matrix decompositions Mathrdquo Numerica Sinica vol 24 pp165ndash176 2002

[24] A P Liao and Y Lei ldquoLeast-squares solution with the mini-mum-norm for the matrix equation (119860119883119861 119866119883119867) = (119862119863)rdquoComputers and Mathematics with Applications vol 50 no 3-4pp 539ndash549 2005

[25] Y H Liu ldquoRanks of least squares solutions of the matrix equa-tion 119860119883119861 = 119862rdquo Computers and Mathematics with Applicationsvol 55 no 6 pp 1270ndash1278 2008

[26] X Sheng and G Chen ldquoA finite iterative method for solving apair of linear matrix equations (119860119883119861 119862119883119863) = (119864 119865)rdquo AppliedMathematics and Computation vol 189 no 2 pp 1350ndash13582007

[27] A G Wu W Liu and G R Duan ldquoOn the conjugate productof complex polynomial matricesrdquo Mathematical and ComputerModelling vol 53 no 9-10 pp 2031ndash2043 2011

[28] A GWu G FengW Liu and G R Duan ldquoThe complete solu-tion to the Sylvester-polynomial-conjugate matrix equationsrdquoMathematical and Computer Modelling vol 53 no 9-10 pp2044ndash2056 2011

[29] A G Wu B Li Y Zhang and G R Duan ldquoFinite iterativesolutions to coupled Sylvester-conjugate matrix equationsrdquoApplied Mathematical Modelling vol 35 no 3 pp 1065ndash10802011

[30] AGWu L Lv andGRDuan ldquoIterative algorithms for solvinga class of complex conjugate and transpose matrix equationsrdquoApplied Mathematics and Computation vol 217 no 21 pp8343ndash8353 2011

[31] X ZhangMatrix Analysis and Application TsinghuaUniversityPress Beijing China 2004

[32] A G Wu L Lv and M Z Hou ldquoFinite iterative algorithms forextended Sylvester-conjugate matrix equationsrdquo Mathematicaland Computer Modelling vol 54 pp 2363ndash2384 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Discrete Mathematics 7

= Retr[1198771198672119860111987511198611+ 1198772

119867

119860211987511198612+ 1198772119861119867

31198751119860119867

3

+ 11987721198614

119867

11987511198604

119867

+

100381710038171003817100381711987721003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

2119875119867

11198751+ 119877119867

2119862111987611198631

+ 1198772

119867

119862211987611198632+ 1198772119863119867

31198761119862119867

3

+ 11987721198634

119867

11987611198624

119867

+

100381710038171003817100381711987721003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

2119876119867

11198761]

= Retr[1198771198672119860111987511198611+ 1198772

119867

119860211987511198612+ 1198603119875119867

11198613119877119867

2

+ 1198604119875119879

11198614119877119867

2+ 119877119867

2119862111987611198631

+ 1198772

119867

119862211987611198632+ 1198623119876119867

11198633119877119867

2

+ 1198624119876119879

11198634119877119867

2+

100381710038171003817100381711987721003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

2(119875119867

11198751+ 119876119867

11198761)]

= Retr[1198771198672119860111987511198611+ 119877119867

2119860211987511198612+ 119877119867

21198603119875119867

11198613

+ 119877119867

21198604119875119879

11198614+ 119877119867

2119862111987611198631

+ 119877119867

2119862211987611198632+ 119877119867

21198623119876119867

11198633

+ 119877119867

21198624119876119879

11198634+

100381710038171003817100381711987721003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

2(119875119867

11198751+ 119876119867

11198761)]

= Retr[1198771198672(119860111987511198611+ 119862111987611198631+ 119860211987511198612

+ 119862211987611198632+ 1198603119875119867

11198613+ 1198623119876119867

11198633

+1198604119875119879

11198614+ 1198624119876119867

11198634) ]

+

100381710038171003817100381711987721003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

2(10038171003817100381710038171198751

1003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2

)

=

100381710038171003817100381711987511003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

2Re tr [119877119867

2(1198771minus 1198772)]

+

100381710038171003817100381711987721003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

2(10038171003817100381710038171198751

1003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2

)

= minus

100381710038171003817100381711987511003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

2(10038171003817100381710038171198772

1003817100381710038171003817

2

)

+

100381710038171003817100381711987721003817100381710038171003817

2

100381710038171003817100381711987711003817100381710038171003817

2(10038171003817100381710038171198751

1003817100381710038171003817

2

+10038171003817100381710038171198761

1003817100381710038171003817

2

) = 0

(21)This implies that (18) is satisfied for 119894 = 1

Assume that (17) and (18) hold for 119894 = 119896 minus 1 fromAlgorithm 3 we have

Re tr (119877119867119896+1

119877119896)

= Re

tr[

[

(119877119896minus

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

times (11986011198751198961198611+ 11986211198761198961198631+ 11986021198751198961198612

+ 11986221198761198961198632+ 1198603119875119867

1198961198613+ 1198623119876119867

1198961198633

+ 1198604119875119879

1198961198614+ 1198624119876119879

1198961198634))

119867

119877119896]

]

Re tr (119877119867119896+1

119877119896)

= Retr (119877119867119896119877119896) minus

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

times tr (119875119867119896119860119867

1119877119896119861119867

1+ 119876119867

119896119862119867

1119877119896119863119867

1

+ 119875119896

119867

119860119867

2119877119896119861119867

2+ 119876119896

119867

119862119867

2119877119896119863119867

2

+ 119875119896119860119867

3119877119896119861119867

3+ 119876119896119862119867

3119877119896119863119867

3

+ 119875119896119860119867

4119877119896119861119867

4+ 119876119896119862119867

4119877119896119863119867

4)

=1003817100381710038171003817119877119896

1003817100381710038171003817

2

minus

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

times Re tr (119875119867119896119860119867

1119877119896119861119867

1+ 119876119867

119896119862119867

1119877119896119863119867

1

+ 119875119896

119867

119860119867

2119877119896119861119867

2+ 119876119896

119867

119862119867

2119877119896119863119867

2

+ 1198613119877119867

1198961198603119875119867

119896+ 1198633119877119867

1198961198623119876119867

119896

+ 1198614119877119879

1198961198604119875119867

119896+ 1198634119877119879

1198961198624119876119867

119896)

=10038171003817100381710038171198771

1003817100381710038171003817

2

minus

100381710038171003817100381711987711003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

times Re tr [119875119867119896

(119860119867

1119877119896119861119867

1+ 119860119867

2119877119896119861119867

2

+ 1198613119877119867

1198961198603+ 1198614119877119879

1198961198604)

+ 119876119867

119896(119862119867

1119877119896119863119867

1+ 119862119867

2119877119896119863119867

2

+ 1198633119877119867

1198961198623+ 1198634119877119879

1198961198624)]

8 Journal of Discrete Mathematics

=1003817100381710038171003817119877119896

1003817100381710038171003817

2

minus

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

times Retr[119875119867119896

(119875119896minus

10038171003817100381710038171198771198961003817100381710038171003817

2

1003817100381710038171003817119877119896minus11003817100381710038171003817

2119875119896minus1

)

+ 119876119867

119896(119876119896minus

10038171003817100381710038171198771198961003817100381710038171003817

2

1003817100381710038171003817119877119896minus11003817100381710038171003817

2119876119896minus1

)]

=1003817100381710038171003817119877119896

1003817100381710038171003817

2

minus

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2(1003817100381710038171003817119875119896

1003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

) = 0

(22)

Thus (17) holds for 119894 = 119896Also from Algorithm 3 we have

Re tr (119875119867119896+1

119875119896+ 119876119867

119896+1119876119896)

= Re

tr [

[

(119860119867

1119877119896+1

119861119867

1+ 119860119867

2119877119896+1

119861119867

2

+ 1198613119877119867

119896+11198603+ 1198614119877119879

119896+11198604

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2119875119896)

119867

119875119896

+ (119862119867

1119877119896+1

119863119867

1+ 119862119867

2119877119896+1

119863119867

2

+ 1198633119877119867

119896+11198623+ 1198634119877119879

119896+11198624

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2119876119896)

119867

119876119896]

]

= Retr[119877119867119896+1

11986011198751198961198611+ 119877119896+1

119867

11986021198751198961198612

+ 119877119896+1

119861119867

3119875119896119860119867

3+ 119877119896+1

1198614

119867

1198751198961198604

119867

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2119875119867

119896119875119896+ 119877119867

119896+111986211198761198961198631

+ 119877119896+1

119867

11986221198761198961198632+ 119877119896+1

119863119867

3119876119896119862119867

3

+ 119877119896+1

1198634

119867

1198761198961198624

119867

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2119876119867

119896119876119896]

= Retr[119877119867119896+1

11986011198751198961198611+ 119877119896+1

119867

11986021198751198961198612

+ 1198603119875119867

1198961198613119877119867

119896+1+ 1198604119875119879

1198961198614119877119867

119896+1

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2119875119867

119896119875119896+ 119877119867

119896+111986211198761198961198631

+ 119877119896+1

119867

11986221198761198961198632+ 1198623119876119867

1198961198633119877119867

119896+1

+ 1198624119876119879

1198961198634119877119867

119896+1+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2119876119867

119896119876119896]

= Retr[119877119867119896+1

(11986011198751198961198611+ 11986211198761198961198631

+ 11986021198751198961198612+ 11986221198761198961198632+ 1198603119875119867

1198961198613

+1198623119876119867

1198961198633+ 1198604119875119879

1198961198614+ 1198624119876119879

1198961198634)

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2(119875119867

119896119875119896+ 119876119867

119896119876119896)]

=

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2Re tr (119877119867

119896+1(119877119896minus 119877119896+1

))

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2(1003817100381710038171003817119875119896

1003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

)

= minus

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2(1003817100381710038171003817119877119896+1

1003817100381710038171003817

2

)

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2(1003817100381710038171003817119875119896

1003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

) = 0

(23)

This implies that (17) and (18) hold for 119894 = 119896Then relations (17) and (18) holds bymathematical induc-

tionStep 2 We want to show that

Re (tr (119877119867119894+119897119877119894)) = 0

Re (tr (119875119867119894+119897119875119894+ 119876119867

119894+119897119876119894)) = 0

(24)

holds for 119897 ge 1 We will prove this conclusion by inductionThe case of 119897 = 1 has been proven in Step 1 Now we assumethat (24) holds for 119897 le 119904 119904 ge 1 The aim is to show that

Re (tr (119877119867119894+119904+1

119877119894)) = 0

Re (tr (119875119867119894+119904+1

119875119894+ 119876119867

119894+119904+1119876119894)) = 0

(25)

First we prove the following

Re (tr (119877119867119904+1

1198770)) = 0

Re (tr (119875119867119904+1

1198750+ 119876119867

119904+11198760)) = 0

(26)

Journal of Discrete Mathematics 9

By using Algorithm 3 from (19) and induction we have

Re tr (119877119867119904+1

1198770)

= Re

tr[

[

(119877119904minus

10038171003817100381710038171198771199041003817100381710038171003817

2

10038171003817100381710038171198751199041003817100381710038171003817

2

+1003817100381710038171003817119876119904

1003817100381710038171003817

2

times (11986011198751199041198611+ 11986211198761199041198631+ 11986021198751199041198612

+ 11986221198761199041198632+ 1198603119875119867

1199041198613+ 1198623119876119867

1199041198633

+ 1198604119875119879

1199041198614+ 1198624119876119879

1199041198634))

119867

1198770]

]

= Re tr (1198771198671199041198770) minus

10038171003817100381710038171198771199041003817100381710038171003817

2

10038171003817100381710038171198751199041003817100381710038171003817

2

+1003817100381710038171003817119876119904

1003817100381710038171003817

2

times tr (119875119867119904119860119867

11198770119861119867

1+ 119876119867

119904119862119867

11198770119863119867

1

+ 119875119904

119867

119860119867

21198770119861119867

2+ 119876119904

119867

119862119867

21198770119863119867

2

+ 119875119904119860119867

31198770119861119867

3+ 119876119904119862119867

31198770119863119867

3

+ 119875119904119860119867

41198770119861119867

4+ 119876119904119862119867

41198770119863119867

4)

Re tr (119877119867119904+1

1198770)

= Re tr (1198771198671199041198770) minus

10038171003817100381710038171198771199041003817100381710038171003817

2

10038171003817100381710038171198751199041003817100381710038171003817

2

+1003817100381710038171003817119876119904

1003817100381710038171003817

2

times Re tr (119875119867119904119860119867

11198770119861119867

1+ 119876119867

119904119862119867

11198770119863119867

1

+ 119875119904

119867

119860119867

21198770119861119867

2+ 119876119904

119867

119862119867

21198770119863119867

2

+ 1198613119877119867

01198603119875119867

119904+ 1198633119877119867

01198623119876119867

119904

+ 1198614119877119879

01198604119875119867

119904+ 1198634119877119879

01198624119876119867

119904)

= minus

10038171003817100381710038171198771199041003817100381710038171003817

2

10038171003817100381710038171198751199041003817100381710038171003817

2

+1003817100381710038171003817119876119904

1003817100381710038171003817

2

times Re tr (119875119867119904

(119860119867

11198770119861119867

1+ 119860119867

21198770119861119867

2

+ 1198613119877119867

01198603+ 1198614119877119879

01198604)

+ 119876119867

119904(119862119867

11198770119863119867

1+ 119862119867

21198770119863119867

2

+1198633119877119867

01198623+ 1198634119877119879

01198624))

= minus

10038171003817100381710038171198771199041003817100381710038171003817

2

10038171003817100381710038171198751199041003817100381710038171003817

2

+1003817100381710038171003817119876119904

1003817100381710038171003817

2Re tr (119875119867

1199041198750+ 119876119867

1199041198760) = 0

Re tr (119875119867119904+1

1198750+ 119876119867

119904+11198760)

= Retr[(119860119867

1119877119904+1

119861119867

1+ 119860119867

2119877119904+1

119861119867

2

+ 1198613119877119867

119904+11198603+ 1198614119877119879

119904+11198604

+

1003817100381710038171003817119877119904+11003817100381710038171003817

2

10038171003817100381710038171198771199041003817100381710038171003817

2119875119904)

119867

1198750

+ (119862119867

1119877119904+1

119863119867

1+ 119862119867

2119877119904+1

119863119867

2

+ 1198633119877119867

119904+11198623+ 1198634119877119879

119896+11198624

+

1003817100381710038171003817119877119904+11003817100381710038171003817

2

10038171003817100381710038171198771199041003817100381710038171003817

2119876119904)

119867

1198760]

]

= Retr[119877119867119904+1

119860111987501198611+ 119877119904+1

119867

119860211987501198612

+ 119877119904+1

119861119867

31198750119860119867

3+ 119877119904+1

1198614

119867

11987501198604

119867

+

1003817100381710038171003817119877119904+11003817100381710038171003817

2

10038171003817100381710038171198771199041003817100381710038171003817

2119875119867

1199041198750+ 119877119867

119904+1119862111987601198631

+ 119877119904+1

119867

119862211987601198632+ 119877119904+1

119863119867

31198760119862119867

3

+ 119877119904+1

1198634

119867

11987601198624

119867

+

1003817100381710038171003817119877119904+11003817100381710038171003817

2

10038171003817100381710038171198771199041003817100381710038171003817

2119876119867

1199041198760]

= Retr[119877119867119904+1

119860111987501198611+ 119877119904+1

119867

119860211987501198612

+ 1198603119875119867

01198613119877119867

119904+1+ 1198604119875119879

01198614119877119867

119904+1

+

1003817100381710038171003817119877119904+11003817100381710038171003817

2

10038171003817100381710038171198771199041003817100381710038171003817

2119875119867

1199041198750+ 119877119867

119904+1119862111987601198631

+ 119877119904+1

119867

119862211987601198632+ 1198623119876119867

01198633119877119867

119904+1

+ 1198624119876119879

01198634119877119867

119904+1+

1003817100381710038171003817119877119904+11003817100381710038171003817

2

10038171003817100381710038171198771199041003817100381710038171003817

2119876119867

1199041198760]

= Retr[119877119867119904+1

(119860111987501198611+ 119862111987601198631+ 119860211987501198612

+ 119862211987601198632+ 1198603119875119867

01198613

+ 1198623119876119867

01198633+ 1198604119875119879

01198614

+ 1198624119876119879

01198634)

+

1003817100381710038171003817119877119904+11003817100381710038171003817

2

10038171003817100381710038171198771199041003817100381710038171003817

2(119875119867

1199041198750+ 119876119867

1199041198760)]

=

100381710038171003817100381711987501003817100381710038171003817

2

+10038171003817100381710038171198760

1003817100381710038171003817

2

100381710038171003817100381711987701003817100381710038171003817

2Re tr (119877119867

119904+1(1198770minus 1198771))= 0

(27)

Then (26) is holds

10 Journal of Discrete Mathematics

From Algorithm 3 we have

Re tr (119875119867119894+119904+1

119875119894+ 119876119867

119894+119904+1119876119894)

= Retr[(119860119867

1119877119894+119904+1

119861119867

1+ 119860119867

2119877119894+119904+1

119861119867

2

+ 1198613119877119867

119894+119904+11198603+ 1198614119877119879

119894+119904+11198604

+

1003817100381710038171003817119877119894+119904+11003817100381710038171003817

2

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2119875119894+119904

)

119867

119875119894

+ (119862119867

1119877119894+119904+1

119863119867

1+ 119862119867

2119877119894+119904+1

119863119867

2

+ 1198633119877119867

119894+119904+11198623+ 1198634119877119879

119894+119904+11198624

+

1003817100381710038171003817119877119894+119904+11003817100381710038171003817

2

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2119876119894+119904

)

119867

119876119894]

]

= Retr[119877119867119894+119904+1

11986011198751198941198611+ 119877119894+119904+1

119867

11986021198751198941198612

+ 119877119894+119904+1

119861119867

3119875119894119860119867

3+ 119877119894+119904+1

1198614

119867

1198751198941198604

119867

+

1003817100381710038171003817119877119894+119904+11003817100381710038171003817

2

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2119875119867

119894+119904119875119894+ 119877119867

119894+119904+111986211198761198941198631

+ 119877119894+119904+1

119867

11986221198761198941198632+ 119877119894+119904+1

119863119867

3119876119894119862119867

3

+ 119877119894+119904+1

1198634

119867

1198761198941198624

119867

+

1003817100381710038171003817119877119894+119904+11003817100381710038171003817

2

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2119876119867

119894+119904119876119894]

= Retr[119877119867119894+119904+1

11986011198751198941198611+ 119877119894+119904+1

119867

11986021198751198941198612

+ 1198603119875119867

1198941198613119877119867

119894+119904+1+ 1198604119875119879

1198941198614119877119867

119894+119904+1

+

1003817100381710038171003817119877119894+119904+11003817100381710038171003817

2

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2119875119867

119894+119904119875119894+ 119877119867

119894+119904+111986211198761198941198631

+ 119877119894+119904+1

119867

11986221198761198941198632+ 1198623119876119867

1198941198633119877119867

119894+119904+1

+ 1198624119876119879

1198961198634119877119867

119894+119904+1+

1003817100381710038171003817119877119894+119904+11003817100381710038171003817

2

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2119876119867

119894+119904119876119894]

= Retr[119877119867119894+119904+1

(11986011198751198941198611+ 11986211198761198941198631

+ 11986021198751198941198612+ 11986221198761198941198632

+ 1198603119875119867

1198941198613+ 1198623119875119867

1198941198633

+ 1198604119875119879

1198941198614+ 1198624119876119879

1198941198634)

+

1003817100381710038171003817119877119894+119904+11003817100381710038171003817

2

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2(119875119867

119894+119904119875119894+ 119876119867

119894+119904119876119894)]

=

10038171003817100381710038171198751198941003817100381710038171003817

2

+1003817100381710038171003817119876119894

1003817100381710038171003817

2

10038171003817100381710038171198771198941003817100381710038171003817

2Re tr (119877119867

119894+119904+1(119877119894minus 119877119894+1

))

=

10038171003817100381710038171198751198941003817100381710038171003817

2

+1003817100381710038171003817119876119894

1003817100381710038171003817

2

10038171003817100381710038171198771198941003817100381710038171003817

2Re tr (119877119867

119894+119904+1119877119894)

(28)

Also from (19) we have

Re tr (119877119867119894+119904+1

119877119894)

= Re

tr[

[

(119877119894+119904

minus

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2

1003817100381710038171003817119875119894+1199041003817100381710038171003817

2

+1003817100381710038171003817119876119894+119904

1003817100381710038171003817

2

times (1198601119875119894+119904

1198611+ 1198621119876119894+119904

1198631

+ 1198602119875119894+119904

1198612+ 1198622119876119894+119904

1198632

+ 1198603119875119867

119894+1199041198613+ 1198623119876119867

119894+1199041198633

+ 1198604119875119879

119894+1199041198614+ 1198624119876119879

119894+1199041198634))

119867

119877119894]

]

= Retr (119877119867119894+119904

119877119894) minus

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2

1003817100381710038171003817119875119894+1199041003817100381710038171003817

2

+1003817100381710038171003817119876119894+119904

1003817100381710038171003817

2

times tr (119875119867119894+119904

119860119867

1119877119894119861119867

1+ 119876119867

119894+119904119862119867

1119877119894119863119867

1

+ 119875119894+119904

119867

119860119867

2119877119894119861119867

2+ 119876119894+119904

119867

119862119867

2119877119894119863119867

2

+ 119875119894+119904

119860119867

3119877119894119861119867

3+ 119876119894+119904

119862119867

3119877119894119863119867

3

+ 119875119894+119904

119860119867

4119877119894119861119867

4+ 119876119894+119904

119862119867

4119877119894119863119867

4)

Re tr (119877119867119894+119904+1

119877119894)

= minus

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2

1003817100381710038171003817119875119894+1199041003817100381710038171003817

2

+1003817100381710038171003817119876119894+119904

1003817100381710038171003817

2

times Retr(119875119867119894+119904

119860119867

1119877119894119861119867

1+ 119876119867

119894+119904119862119867

1119877119894119863119867

1

+ 119875119894+119904

119867

119860119867

2119877119894119861119867

2+ 119876119894+119904

119867

119862119867

2119877119894119863119867

2

+ 1198613119877119867

1198941198603119875119867

119894+119904+ 1198633119877119867

1198941198623119876119867

119894+119904

+ 1198614119877119879

1198941198604119875119867

119894+119904+ 1198634119877119879

1198941198624119876119867

119894+119904)

Journal of Discrete Mathematics 11

= minus

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2

1003817100381710038171003817119875119894+1199041003817100381710038171003817

2

+1003817100381710038171003817119876119894+119904

1003817100381710038171003817

2

times Re tr [119875119867119894+119904

(119860119867

1119877119894119861119867

1+ 119860119867

2119877119894119861119867

2

+ 1198613119877119867

1198941198603+ 1198614119877119879

1198941198604)

+ 119876119867

119894+119904(119862119867

1119877119894119863119867

1+ 119862119867

2119877119894119863119867

2

+ 1198633119877119867

1198941198623+ 1198634119877119879

1198941198624)]

= minus

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2

1003817100381710038171003817119875119894+1199041003817100381710038171003817

2

+1003817100381710038171003817119876119894+119904

1003817100381710038171003817

2

times Re tr[119875119867119894+119904

(119875119894minus

10038171003817100381710038171198771198941003817100381710038171003817

2

1003817100381710038171003817119877119894minus11003817100381710038171003817

2119875119894minus1

)

+ 119876119867

119894+119904(119876119894minus

10038171003817100381710038171198771198941003817100381710038171003817

2

1003817100381710038171003817119877119894minus11003817100381710038171003817

2119876119894minus1

)]

=

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2

1003817100381710038171003817119875119894+1199041003817100381710038171003817

2

+1003817100381710038171003817119876119894+119904

1003817100381710038171003817

2

10038171003817100381710038171198771198941003817100381710038171003817

2

1003817100381710038171003817119877119894minus11003817100381710038171003817

2

times Re tr (119875119867119894+119904

119875119894minus1

+ 119876119867

119894+119904119876119894minus1

)

(29)

Repeating (28) and (29) one can easily obtain for certain 120572

and 120573

tr (119875119867119894+119904+1

119875119894+ 119876119867

119894+119904+1119876119894) = 120572 [tr (119875119867

119904+11198751+ 119876119867

119904+11198761)]

tr (119877119867119894+119904+1

119877119894) = 120573 [tr (119877119867

119904+11198771)]

(30)

Combining these two relations with (26) implies that (24)holds for 119897 = 119904 + 1 From Steps 1 and 2 the conclusion holdsby the principle of inductionWith the above two lemmas wehave the following theorem

Theorem 6 (see [32]) If the matrix equation (1) is consistentthen a solution can be obtained within finite iteration steps byusing Algorithm 3 for any initial matrices 119881

11198821

4 Numerical Example

In this section we present numerical example to illustrate theapplication of our proposed methods

Example 7 In this example we illustrate our theoreticalresults of Algorithm 3 for solving the system of matrixequation

11986011198811198611+ 11986211198821198631+ 11986021198811198612+ 11986221198821198632+ 11986031198811198671198613

+ 11986231198821198671198633+ 11986041198811198791198614+ 11986241198821198791198634= 119864

(31)

Because of the influence of the error of calculation theresidual 119877(119896) is usually unequal to zero in this process ofthe iteration We regard the matrix 119877(119896) as a zero matrix if119877(119896) lt 10

minus10Given

1198601= [

2 + 3119894 minus119894 1 + 119894

5 1 + 2119894 minus3]

1198602= [

2 + 3119894 minus119894 1 + 119894

5 1 + 2119894 minus3]

1198603= [

0 2 minus 119894 119894

minus1 + 3119894 2 0] 119860

4=[

0 1 minus 3119894 1 + 119894

0 4 + 119894 minus3119894]

1198621= [

1 + 2119894 3 minus 119894 4

minus119894 2119894 minus3] 119862

2=[

3 + 2119894 0 1 + 119894

0 4119894 1 minus 2119894]

1198623= [

1 minus 3119894 2119894 minus3119894

1 2 + 3119894 4119894] 119862

4=[

1 minus 2119894 0 2

3 minus 119894 1 + 119894 minus1]

1198611= [

[

4 + 119894 minus119894

0 1 minus 119894

4119894 2 + 2119894

]

]

1198612= [

[

0 119894

1 + 119894 0

minus1 minus 119894 3119894

]

]

1198613= [

[

0 1

minus3119894 4 + 119894

5 1 + 2119894

]

]

1198614=[

[

3 + 119894 minus1 minus 119894

0 2 minus 119894

minus1 + 119894 2

]

]

1198631= [

[

0 0

1 minus 3119894 minus119894

2119894 minus3119894

]

]

1198632= [

[

0 119894

1 + 119894 0

minus1 minus 119894 3119894

]

]

1198633= [

[

0 1

minus3119894 4 + 119894

5 1 + 2119894

]

]

1198634= [

[

3119894 minus2 + 119894

0 119894

minus2119894 minus4119894

]

]

119864 = [42 + 55119894 115 + 25119894

minus38 minus 119894 132 + 44119894]

(32)

Taking 1198811= [0 0

0 0

0 0

] and 1198821= [0 0

0 0] we apply Algorithm 3

to compute 119881119896119882119896

And iterating 42 steps we get

119881

=[

[

00126 + 18415119894 00827 + 06381119894 11221 minus 09428119894

minus06903 + 10185119894 18818 + 10203119894 09208 + 04569119894

05344 minus 04909119894 09280 + 07169119894 04872 minus 02734119894

]

]

119882

=[

[

04218 minus 09710119894 01763 + 05183119894 11331 minus 00432119894

06273 minus 01216119894 minus03902 + 04313119894 06240 + 09828119894

minus07011 minus 03418119894 03695 + 17627119894 minus03032 + 07073119894

]

]

(33)

which satisfy the matrix equation

11986011198811198611+ 11986211198821198631+ 11986021198811198612+ 11986221198821198632+ 11986031198811198671198613

+ 11986231198821198671198633+ 11986041198811198791198614+ 11986241198821198791198634= 119864

(34)

12 Journal of Discrete Mathematics

log10 rk

4

2

0

minus2

minus4

minus6

minus8

minus100 5 10 15 20 25 30 35 40 45

k (iteration number)

Figure 1The relation between the number of iterations and residualfor the example

With the corresponding residual100381710038171003817100381711987742

1003817100381710038171003817

=10038171003817100381710038171003817119864 minus 119860

1119881421198611minus 1198621119882421198631minus 1198602119881421198612minus 1198622119882421198632

minus 1198603119881119867

421198613minus 1198623119882119867

421198633minus 1198604119881119879

421198614minus 1198624119882119879

421198634

10038171003817100381710038171003817

= 66115 times 10minus11

(35)

5 Conclusions

The above Figure 1 shows the convergence curve for theresidual function 119877(119896) In this paper an iterative algorithmconstructed to solve a complex matrix equation with conju-gate and transpose of two unknowns of the form 119860

11198811198611+

11986211198821198631+11986021198811198612+11986221198821198632+11986031198811198671198613+11986231198821198671198633+11986041198811198791198614+

11986241198821198791198634

= 119864 is presented We proved that the iterativealgorithms always converge to the solution for any initialmatrices We stated and proved some lemmas and theoremswhere the solutions are obtained The proposed methodis illustrated by numerical example where the obtainednumerical results show that our technique is very neat andefficient

References

[1] F Ding P X Liu and J Ding ldquoIterative solutions of thegeneralized Sylvester matrix equations by using the hierarchicalidentification principlerdquo Applied Mathematics and Computa-tion vol 197 no 1 pp 41ndash50 2008

[2] F Ding and T Chen ldquoGradient based iterative algorithmsfor solving a class of matrix equationsrdquo IEEE Transactions onAutomatic Control vol 50 no 8 pp 1216ndash1221 2005

[3] FDing andTChen ldquoHierarchical gradient-based identificationof multivariable discrete-time systemsrdquo Automatica vol 41 no2 pp 315ndash325 2005

[4] F Ding and T Chen ldquoIterative least-squares solutions of cou-pled Sylvester matrix equationsrdquo Systems and Control Lettersvol 54 no 2 pp 95ndash107 2005

[5] F Ding and T Chen ldquoOn iterative solutions of general coupledmatrix equationsrdquo SIAM Journal on Control and Optimizationvol 44 no 6 pp 2269ndash2284 2006

[6] F Ding and T Chen ldquoHierarchical least squares identificationmethods for multivariable systemsrdquo IEEE Transactions onAutomatic Control vol 50 no 3 pp 397ndash402 2005

[7] F Ding and T Chen ldquoPerformance analysis ofmulti-innovationgradient type identification methodsrdquo Automatica vol 43 no1 pp 1ndash14 2007

[8] AGWuX ZengG RDuan andW JWu ldquoIterative solutionsto the extended Sylvester-conjugate matrix equationsrdquo AppliedMathematics and Computation vol 217 no 1 pp 130ndash142 2010

[9] B Zhou G R Duan and Z Y Li ldquoGradient based iterativealgorithm for solving coupled matrix equationsrdquo Systems andControl Letters vol 58 no 5 pp 327ndash333 2009

[10] J H Bevis F J Hall and R E Hartwing ldquoConsimilarity andthe matrix equation119860119883minus119883119861 = 119862rdquo in Current Trends inMatrixTheory pp 51ndash64 North-Holland New York NY USA 1987

[11] R A Horn and C R Johnson Matrix Analysis CambridgeUniversity Press Cambridge UK 1990

[12] H Liping ldquoConsimilarity of quaternion matrices and complexmatricesrdquo Linear Algebra and Its Applications vol 331 no 1ndash3pp 21ndash30 2001

[13] T Jiang X Cheng and L Chen ldquoAn algebraic relation betweenconsimilarity and similarity of complex matrices and its appli-cationsrdquo Journal of Physics A vol 39 no 29 pp 9215ndash92222006

[14] J H Bevis F J Hall and R E Hartwig ldquoThe matrix equation119860119883 minus 119883119861 = 119862 and its special casesrdquo SIAM Journal on MatrixAnalysis and Applications vol 9 no 3 pp 348ndash359 1988

[15] A G Wu G R Duan and H H Yu ldquoOn solutions of thematrix equations 119883119865 minus 119860119883 = 119862 and 119883119865 minus 119860119883 = 119862rdquo AppliedMathematics and Computation vol 183 no 2 pp 932ndash9412006

[16] A Navarra P L Odell and D M Young ldquoRepresentationof the general common solution to the matrix equations11986011198831198611= 1198621 11986021198831198612= 1198622with applicationsrdquo Computers and

Mathematics with Applications vol 41 no 7-8 pp 929ndash9352001

[17] J W Van der Woude ldquoOn the existence of a common solutionX to the matrix equations 119860

119894119883119861119895= 119862 (i j) E Irdquo Linear Algebra

and Its Applications vol 375 no 1ndash3 pp 135ndash145 2003[18] P Bhimasankaram ldquoCommon solutions to the linear matrix

equations 119860119883 = 119862119883119861 = 119863 and EXF = Grdquo Sankhya Series Avol 38 pp 404ndash409 1976

[19] S Kumar Mitra ldquoThe matrix equations 119860119883 = 119862119883119861 = 119863rdquoLinear Algebra and Its Applications vol 59 pp 171ndash181 1984

[20] S K Mitra ldquoA pair of simultaneous linear matrix equations11986011198831198611

= 1198621 11986021198831198612

= 1198622and a matrix programming

problemrdquo Linear Algebra and Its Applications vol 131 pp 107ndash123 1990

[21] M A Ramadan M A Abdel Naby and A M E BayoumildquoOn the explicit solutions of forms of the Sylvester and theYakubovich matrix equationsrdquo Mathematical and ComputerModelling vol 50 no 9-10 pp 1400ndash1408 2009

[22] K W E Chu ldquoSingular value and generalized singular valuedecompositions and the solution of linear matrix equationsrdquoLinear Algebra and Its Applications vol 88-89 pp 83ndash98 1987

Journal of Discrete Mathematics 13

[23] Y X Yuan ldquoThe optimal solution of linear matrix equationby matrix decompositions Mathrdquo Numerica Sinica vol 24 pp165ndash176 2002

[24] A P Liao and Y Lei ldquoLeast-squares solution with the mini-mum-norm for the matrix equation (119860119883119861 119866119883119867) = (119862119863)rdquoComputers and Mathematics with Applications vol 50 no 3-4pp 539ndash549 2005

[25] Y H Liu ldquoRanks of least squares solutions of the matrix equa-tion 119860119883119861 = 119862rdquo Computers and Mathematics with Applicationsvol 55 no 6 pp 1270ndash1278 2008

[26] X Sheng and G Chen ldquoA finite iterative method for solving apair of linear matrix equations (119860119883119861 119862119883119863) = (119864 119865)rdquo AppliedMathematics and Computation vol 189 no 2 pp 1350ndash13582007

[27] A G Wu W Liu and G R Duan ldquoOn the conjugate productof complex polynomial matricesrdquo Mathematical and ComputerModelling vol 53 no 9-10 pp 2031ndash2043 2011

[28] A GWu G FengW Liu and G R Duan ldquoThe complete solu-tion to the Sylvester-polynomial-conjugate matrix equationsrdquoMathematical and Computer Modelling vol 53 no 9-10 pp2044ndash2056 2011

[29] A G Wu B Li Y Zhang and G R Duan ldquoFinite iterativesolutions to coupled Sylvester-conjugate matrix equationsrdquoApplied Mathematical Modelling vol 35 no 3 pp 1065ndash10802011

[30] AGWu L Lv andGRDuan ldquoIterative algorithms for solvinga class of complex conjugate and transpose matrix equationsrdquoApplied Mathematics and Computation vol 217 no 21 pp8343ndash8353 2011

[31] X ZhangMatrix Analysis and Application TsinghuaUniversityPress Beijing China 2004

[32] A G Wu L Lv and M Z Hou ldquoFinite iterative algorithms forextended Sylvester-conjugate matrix equationsrdquo Mathematicaland Computer Modelling vol 54 pp 2363ndash2384 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Journal of Discrete Mathematics

=1003817100381710038171003817119877119896

1003817100381710038171003817

2

minus

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

times Retr[119875119867119896

(119875119896minus

10038171003817100381710038171198771198961003817100381710038171003817

2

1003817100381710038171003817119877119896minus11003817100381710038171003817

2119875119896minus1

)

+ 119876119867

119896(119876119896minus

10038171003817100381710038171198771198961003817100381710038171003817

2

1003817100381710038171003817119877119896minus11003817100381710038171003817

2119876119896minus1

)]

=1003817100381710038171003817119877119896

1003817100381710038171003817

2

minus

10038171003817100381710038171198771198961003817100381710038171003817

2

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2(1003817100381710038171003817119875119896

1003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

) = 0

(22)

Thus (17) holds for 119894 = 119896Also from Algorithm 3 we have

Re tr (119875119867119896+1

119875119896+ 119876119867

119896+1119876119896)

= Re

tr [

[

(119860119867

1119877119896+1

119861119867

1+ 119860119867

2119877119896+1

119861119867

2

+ 1198613119877119867

119896+11198603+ 1198614119877119879

119896+11198604

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2119875119896)

119867

119875119896

+ (119862119867

1119877119896+1

119863119867

1+ 119862119867

2119877119896+1

119863119867

2

+ 1198633119877119867

119896+11198623+ 1198634119877119879

119896+11198624

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2119876119896)

119867

119876119896]

]

= Retr[119877119867119896+1

11986011198751198961198611+ 119877119896+1

119867

11986021198751198961198612

+ 119877119896+1

119861119867

3119875119896119860119867

3+ 119877119896+1

1198614

119867

1198751198961198604

119867

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2119875119867

119896119875119896+ 119877119867

119896+111986211198761198961198631

+ 119877119896+1

119867

11986221198761198961198632+ 119877119896+1

119863119867

3119876119896119862119867

3

+ 119877119896+1

1198634

119867

1198761198961198624

119867

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2119876119867

119896119876119896]

= Retr[119877119867119896+1

11986011198751198961198611+ 119877119896+1

119867

11986021198751198961198612

+ 1198603119875119867

1198961198613119877119867

119896+1+ 1198604119875119879

1198961198614119877119867

119896+1

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2119875119867

119896119875119896+ 119877119867

119896+111986211198761198961198631

+ 119877119896+1

119867

11986221198761198961198632+ 1198623119876119867

1198961198633119877119867

119896+1

+ 1198624119876119879

1198961198634119877119867

119896+1+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2119876119867

119896119876119896]

= Retr[119877119867119896+1

(11986011198751198961198611+ 11986211198761198961198631

+ 11986021198751198961198612+ 11986221198761198961198632+ 1198603119875119867

1198961198613

+1198623119876119867

1198961198633+ 1198604119875119879

1198961198614+ 1198624119876119879

1198961198634)

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2(119875119867

119896119875119896+ 119876119867

119896119876119896)]

=

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2Re tr (119877119867

119896+1(119877119896minus 119877119896+1

))

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2(1003817100381710038171003817119875119896

1003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

)

= minus

10038171003817100381710038171198751198961003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2(1003817100381710038171003817119877119896+1

1003817100381710038171003817

2

)

+

1003817100381710038171003817119877119896+11003817100381710038171003817

2

10038171003817100381710038171198771198961003817100381710038171003817

2(1003817100381710038171003817119875119896

1003817100381710038171003817

2

+1003817100381710038171003817119876119896

1003817100381710038171003817

2

) = 0

(23)

This implies that (17) and (18) hold for 119894 = 119896Then relations (17) and (18) holds bymathematical induc-

tionStep 2 We want to show that

Re (tr (119877119867119894+119897119877119894)) = 0

Re (tr (119875119867119894+119897119875119894+ 119876119867

119894+119897119876119894)) = 0

(24)

holds for 119897 ge 1 We will prove this conclusion by inductionThe case of 119897 = 1 has been proven in Step 1 Now we assumethat (24) holds for 119897 le 119904 119904 ge 1 The aim is to show that

Re (tr (119877119867119894+119904+1

119877119894)) = 0

Re (tr (119875119867119894+119904+1

119875119894+ 119876119867

119894+119904+1119876119894)) = 0

(25)

First we prove the following

Re (tr (119877119867119904+1

1198770)) = 0

Re (tr (119875119867119904+1

1198750+ 119876119867

119904+11198760)) = 0

(26)

Journal of Discrete Mathematics 9

By using Algorithm 3 from (19) and induction we have

Re tr (119877119867119904+1

1198770)

= Re

tr[

[

(119877119904minus

10038171003817100381710038171198771199041003817100381710038171003817

2

10038171003817100381710038171198751199041003817100381710038171003817

2

+1003817100381710038171003817119876119904

1003817100381710038171003817

2

times (11986011198751199041198611+ 11986211198761199041198631+ 11986021198751199041198612

+ 11986221198761199041198632+ 1198603119875119867

1199041198613+ 1198623119876119867

1199041198633

+ 1198604119875119879

1199041198614+ 1198624119876119879

1199041198634))

119867

1198770]

]

= Re tr (1198771198671199041198770) minus

10038171003817100381710038171198771199041003817100381710038171003817

2

10038171003817100381710038171198751199041003817100381710038171003817

2

+1003817100381710038171003817119876119904

1003817100381710038171003817

2

times tr (119875119867119904119860119867

11198770119861119867

1+ 119876119867

119904119862119867

11198770119863119867

1

+ 119875119904

119867

119860119867

21198770119861119867

2+ 119876119904

119867

119862119867

21198770119863119867

2

+ 119875119904119860119867

31198770119861119867

3+ 119876119904119862119867

31198770119863119867

3

+ 119875119904119860119867

41198770119861119867

4+ 119876119904119862119867

41198770119863119867

4)

Re tr (119877119867119904+1

1198770)

= Re tr (1198771198671199041198770) minus

10038171003817100381710038171198771199041003817100381710038171003817

2

10038171003817100381710038171198751199041003817100381710038171003817

2

+1003817100381710038171003817119876119904

1003817100381710038171003817

2

times Re tr (119875119867119904119860119867

11198770119861119867

1+ 119876119867

119904119862119867

11198770119863119867

1

+ 119875119904

119867

119860119867

21198770119861119867

2+ 119876119904

119867

119862119867

21198770119863119867

2

+ 1198613119877119867

01198603119875119867

119904+ 1198633119877119867

01198623119876119867

119904

+ 1198614119877119879

01198604119875119867

119904+ 1198634119877119879

01198624119876119867

119904)

= minus

10038171003817100381710038171198771199041003817100381710038171003817

2

10038171003817100381710038171198751199041003817100381710038171003817

2

+1003817100381710038171003817119876119904

1003817100381710038171003817

2

times Re tr (119875119867119904

(119860119867

11198770119861119867

1+ 119860119867

21198770119861119867

2

+ 1198613119877119867

01198603+ 1198614119877119879

01198604)

+ 119876119867

119904(119862119867

11198770119863119867

1+ 119862119867

21198770119863119867

2

+1198633119877119867

01198623+ 1198634119877119879

01198624))

= minus

10038171003817100381710038171198771199041003817100381710038171003817

2

10038171003817100381710038171198751199041003817100381710038171003817

2

+1003817100381710038171003817119876119904

1003817100381710038171003817

2Re tr (119875119867

1199041198750+ 119876119867

1199041198760) = 0

Re tr (119875119867119904+1

1198750+ 119876119867

119904+11198760)

= Retr[(119860119867

1119877119904+1

119861119867

1+ 119860119867

2119877119904+1

119861119867

2

+ 1198613119877119867

119904+11198603+ 1198614119877119879

119904+11198604

+

1003817100381710038171003817119877119904+11003817100381710038171003817

2

10038171003817100381710038171198771199041003817100381710038171003817

2119875119904)

119867

1198750

+ (119862119867

1119877119904+1

119863119867

1+ 119862119867

2119877119904+1

119863119867

2

+ 1198633119877119867

119904+11198623+ 1198634119877119879

119896+11198624

+

1003817100381710038171003817119877119904+11003817100381710038171003817

2

10038171003817100381710038171198771199041003817100381710038171003817

2119876119904)

119867

1198760]

]

= Retr[119877119867119904+1

119860111987501198611+ 119877119904+1

119867

119860211987501198612

+ 119877119904+1

119861119867

31198750119860119867

3+ 119877119904+1

1198614

119867

11987501198604

119867

+

1003817100381710038171003817119877119904+11003817100381710038171003817

2

10038171003817100381710038171198771199041003817100381710038171003817

2119875119867

1199041198750+ 119877119867

119904+1119862111987601198631

+ 119877119904+1

119867

119862211987601198632+ 119877119904+1

119863119867

31198760119862119867

3

+ 119877119904+1

1198634

119867

11987601198624

119867

+

1003817100381710038171003817119877119904+11003817100381710038171003817

2

10038171003817100381710038171198771199041003817100381710038171003817

2119876119867

1199041198760]

= Retr[119877119867119904+1

119860111987501198611+ 119877119904+1

119867

119860211987501198612

+ 1198603119875119867

01198613119877119867

119904+1+ 1198604119875119879

01198614119877119867

119904+1

+

1003817100381710038171003817119877119904+11003817100381710038171003817

2

10038171003817100381710038171198771199041003817100381710038171003817

2119875119867

1199041198750+ 119877119867

119904+1119862111987601198631

+ 119877119904+1

119867

119862211987601198632+ 1198623119876119867

01198633119877119867

119904+1

+ 1198624119876119879

01198634119877119867

119904+1+

1003817100381710038171003817119877119904+11003817100381710038171003817

2

10038171003817100381710038171198771199041003817100381710038171003817

2119876119867

1199041198760]

= Retr[119877119867119904+1

(119860111987501198611+ 119862111987601198631+ 119860211987501198612

+ 119862211987601198632+ 1198603119875119867

01198613

+ 1198623119876119867

01198633+ 1198604119875119879

01198614

+ 1198624119876119879

01198634)

+

1003817100381710038171003817119877119904+11003817100381710038171003817

2

10038171003817100381710038171198771199041003817100381710038171003817

2(119875119867

1199041198750+ 119876119867

1199041198760)]

=

100381710038171003817100381711987501003817100381710038171003817

2

+10038171003817100381710038171198760

1003817100381710038171003817

2

100381710038171003817100381711987701003817100381710038171003817

2Re tr (119877119867

119904+1(1198770minus 1198771))= 0

(27)

Then (26) is holds

10 Journal of Discrete Mathematics

From Algorithm 3 we have

Re tr (119875119867119894+119904+1

119875119894+ 119876119867

119894+119904+1119876119894)

= Retr[(119860119867

1119877119894+119904+1

119861119867

1+ 119860119867

2119877119894+119904+1

119861119867

2

+ 1198613119877119867

119894+119904+11198603+ 1198614119877119879

119894+119904+11198604

+

1003817100381710038171003817119877119894+119904+11003817100381710038171003817

2

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2119875119894+119904

)

119867

119875119894

+ (119862119867

1119877119894+119904+1

119863119867

1+ 119862119867

2119877119894+119904+1

119863119867

2

+ 1198633119877119867

119894+119904+11198623+ 1198634119877119879

119894+119904+11198624

+

1003817100381710038171003817119877119894+119904+11003817100381710038171003817

2

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2119876119894+119904

)

119867

119876119894]

]

= Retr[119877119867119894+119904+1

11986011198751198941198611+ 119877119894+119904+1

119867

11986021198751198941198612

+ 119877119894+119904+1

119861119867

3119875119894119860119867

3+ 119877119894+119904+1

1198614

119867

1198751198941198604

119867

+

1003817100381710038171003817119877119894+119904+11003817100381710038171003817

2

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2119875119867

119894+119904119875119894+ 119877119867

119894+119904+111986211198761198941198631

+ 119877119894+119904+1

119867

11986221198761198941198632+ 119877119894+119904+1

119863119867

3119876119894119862119867

3

+ 119877119894+119904+1

1198634

119867

1198761198941198624

119867

+

1003817100381710038171003817119877119894+119904+11003817100381710038171003817

2

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2119876119867

119894+119904119876119894]

= Retr[119877119867119894+119904+1

11986011198751198941198611+ 119877119894+119904+1

119867

11986021198751198941198612

+ 1198603119875119867

1198941198613119877119867

119894+119904+1+ 1198604119875119879

1198941198614119877119867

119894+119904+1

+

1003817100381710038171003817119877119894+119904+11003817100381710038171003817

2

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2119875119867

119894+119904119875119894+ 119877119867

119894+119904+111986211198761198941198631

+ 119877119894+119904+1

119867

11986221198761198941198632+ 1198623119876119867

1198941198633119877119867

119894+119904+1

+ 1198624119876119879

1198961198634119877119867

119894+119904+1+

1003817100381710038171003817119877119894+119904+11003817100381710038171003817

2

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2119876119867

119894+119904119876119894]

= Retr[119877119867119894+119904+1

(11986011198751198941198611+ 11986211198761198941198631

+ 11986021198751198941198612+ 11986221198761198941198632

+ 1198603119875119867

1198941198613+ 1198623119875119867

1198941198633

+ 1198604119875119879

1198941198614+ 1198624119876119879

1198941198634)

+

1003817100381710038171003817119877119894+119904+11003817100381710038171003817

2

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2(119875119867

119894+119904119875119894+ 119876119867

119894+119904119876119894)]

=

10038171003817100381710038171198751198941003817100381710038171003817

2

+1003817100381710038171003817119876119894

1003817100381710038171003817

2

10038171003817100381710038171198771198941003817100381710038171003817

2Re tr (119877119867

119894+119904+1(119877119894minus 119877119894+1

))

=

10038171003817100381710038171198751198941003817100381710038171003817

2

+1003817100381710038171003817119876119894

1003817100381710038171003817

2

10038171003817100381710038171198771198941003817100381710038171003817

2Re tr (119877119867

119894+119904+1119877119894)

(28)

Also from (19) we have

Re tr (119877119867119894+119904+1

119877119894)

= Re

tr[

[

(119877119894+119904

minus

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2

1003817100381710038171003817119875119894+1199041003817100381710038171003817

2

+1003817100381710038171003817119876119894+119904

1003817100381710038171003817

2

times (1198601119875119894+119904

1198611+ 1198621119876119894+119904

1198631

+ 1198602119875119894+119904

1198612+ 1198622119876119894+119904

1198632

+ 1198603119875119867

119894+1199041198613+ 1198623119876119867

119894+1199041198633

+ 1198604119875119879

119894+1199041198614+ 1198624119876119879

119894+1199041198634))

119867

119877119894]

]

= Retr (119877119867119894+119904

119877119894) minus

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2

1003817100381710038171003817119875119894+1199041003817100381710038171003817

2

+1003817100381710038171003817119876119894+119904

1003817100381710038171003817

2

times tr (119875119867119894+119904

119860119867

1119877119894119861119867

1+ 119876119867

119894+119904119862119867

1119877119894119863119867

1

+ 119875119894+119904

119867

119860119867

2119877119894119861119867

2+ 119876119894+119904

119867

119862119867

2119877119894119863119867

2

+ 119875119894+119904

119860119867

3119877119894119861119867

3+ 119876119894+119904

119862119867

3119877119894119863119867

3

+ 119875119894+119904

119860119867

4119877119894119861119867

4+ 119876119894+119904

119862119867

4119877119894119863119867

4)

Re tr (119877119867119894+119904+1

119877119894)

= minus

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2

1003817100381710038171003817119875119894+1199041003817100381710038171003817

2

+1003817100381710038171003817119876119894+119904

1003817100381710038171003817

2

times Retr(119875119867119894+119904

119860119867

1119877119894119861119867

1+ 119876119867

119894+119904119862119867

1119877119894119863119867

1

+ 119875119894+119904

119867

119860119867

2119877119894119861119867

2+ 119876119894+119904

119867

119862119867

2119877119894119863119867

2

+ 1198613119877119867

1198941198603119875119867

119894+119904+ 1198633119877119867

1198941198623119876119867

119894+119904

+ 1198614119877119879

1198941198604119875119867

119894+119904+ 1198634119877119879

1198941198624119876119867

119894+119904)

Journal of Discrete Mathematics 11

= minus

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2

1003817100381710038171003817119875119894+1199041003817100381710038171003817

2

+1003817100381710038171003817119876119894+119904

1003817100381710038171003817

2

times Re tr [119875119867119894+119904

(119860119867

1119877119894119861119867

1+ 119860119867

2119877119894119861119867

2

+ 1198613119877119867

1198941198603+ 1198614119877119879

1198941198604)

+ 119876119867

119894+119904(119862119867

1119877119894119863119867

1+ 119862119867

2119877119894119863119867

2

+ 1198633119877119867

1198941198623+ 1198634119877119879

1198941198624)]

= minus

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2

1003817100381710038171003817119875119894+1199041003817100381710038171003817

2

+1003817100381710038171003817119876119894+119904

1003817100381710038171003817

2

times Re tr[119875119867119894+119904

(119875119894minus

10038171003817100381710038171198771198941003817100381710038171003817

2

1003817100381710038171003817119877119894minus11003817100381710038171003817

2119875119894minus1

)

+ 119876119867

119894+119904(119876119894minus

10038171003817100381710038171198771198941003817100381710038171003817

2

1003817100381710038171003817119877119894minus11003817100381710038171003817

2119876119894minus1

)]

=

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2

1003817100381710038171003817119875119894+1199041003817100381710038171003817

2

+1003817100381710038171003817119876119894+119904

1003817100381710038171003817

2

10038171003817100381710038171198771198941003817100381710038171003817

2

1003817100381710038171003817119877119894minus11003817100381710038171003817

2

times Re tr (119875119867119894+119904

119875119894minus1

+ 119876119867

119894+119904119876119894minus1

)

(29)

Repeating (28) and (29) one can easily obtain for certain 120572

and 120573

tr (119875119867119894+119904+1

119875119894+ 119876119867

119894+119904+1119876119894) = 120572 [tr (119875119867

119904+11198751+ 119876119867

119904+11198761)]

tr (119877119867119894+119904+1

119877119894) = 120573 [tr (119877119867

119904+11198771)]

(30)

Combining these two relations with (26) implies that (24)holds for 119897 = 119904 + 1 From Steps 1 and 2 the conclusion holdsby the principle of inductionWith the above two lemmas wehave the following theorem

Theorem 6 (see [32]) If the matrix equation (1) is consistentthen a solution can be obtained within finite iteration steps byusing Algorithm 3 for any initial matrices 119881

11198821

4 Numerical Example

In this section we present numerical example to illustrate theapplication of our proposed methods

Example 7 In this example we illustrate our theoreticalresults of Algorithm 3 for solving the system of matrixequation

11986011198811198611+ 11986211198821198631+ 11986021198811198612+ 11986221198821198632+ 11986031198811198671198613

+ 11986231198821198671198633+ 11986041198811198791198614+ 11986241198821198791198634= 119864

(31)

Because of the influence of the error of calculation theresidual 119877(119896) is usually unequal to zero in this process ofthe iteration We regard the matrix 119877(119896) as a zero matrix if119877(119896) lt 10

minus10Given

1198601= [

2 + 3119894 minus119894 1 + 119894

5 1 + 2119894 minus3]

1198602= [

2 + 3119894 minus119894 1 + 119894

5 1 + 2119894 minus3]

1198603= [

0 2 minus 119894 119894

minus1 + 3119894 2 0] 119860

4=[

0 1 minus 3119894 1 + 119894

0 4 + 119894 minus3119894]

1198621= [

1 + 2119894 3 minus 119894 4

minus119894 2119894 minus3] 119862

2=[

3 + 2119894 0 1 + 119894

0 4119894 1 minus 2119894]

1198623= [

1 minus 3119894 2119894 minus3119894

1 2 + 3119894 4119894] 119862

4=[

1 minus 2119894 0 2

3 minus 119894 1 + 119894 minus1]

1198611= [

[

4 + 119894 minus119894

0 1 minus 119894

4119894 2 + 2119894

]

]

1198612= [

[

0 119894

1 + 119894 0

minus1 minus 119894 3119894

]

]

1198613= [

[

0 1

minus3119894 4 + 119894

5 1 + 2119894

]

]

1198614=[

[

3 + 119894 minus1 minus 119894

0 2 minus 119894

minus1 + 119894 2

]

]

1198631= [

[

0 0

1 minus 3119894 minus119894

2119894 minus3119894

]

]

1198632= [

[

0 119894

1 + 119894 0

minus1 minus 119894 3119894

]

]

1198633= [

[

0 1

minus3119894 4 + 119894

5 1 + 2119894

]

]

1198634= [

[

3119894 minus2 + 119894

0 119894

minus2119894 minus4119894

]

]

119864 = [42 + 55119894 115 + 25119894

minus38 minus 119894 132 + 44119894]

(32)

Taking 1198811= [0 0

0 0

0 0

] and 1198821= [0 0

0 0] we apply Algorithm 3

to compute 119881119896119882119896

And iterating 42 steps we get

119881

=[

[

00126 + 18415119894 00827 + 06381119894 11221 minus 09428119894

minus06903 + 10185119894 18818 + 10203119894 09208 + 04569119894

05344 minus 04909119894 09280 + 07169119894 04872 minus 02734119894

]

]

119882

=[

[

04218 minus 09710119894 01763 + 05183119894 11331 minus 00432119894

06273 minus 01216119894 minus03902 + 04313119894 06240 + 09828119894

minus07011 minus 03418119894 03695 + 17627119894 minus03032 + 07073119894

]

]

(33)

which satisfy the matrix equation

11986011198811198611+ 11986211198821198631+ 11986021198811198612+ 11986221198821198632+ 11986031198811198671198613

+ 11986231198821198671198633+ 11986041198811198791198614+ 11986241198821198791198634= 119864

(34)

12 Journal of Discrete Mathematics

log10 rk

4

2

0

minus2

minus4

minus6

minus8

minus100 5 10 15 20 25 30 35 40 45

k (iteration number)

Figure 1The relation between the number of iterations and residualfor the example

With the corresponding residual100381710038171003817100381711987742

1003817100381710038171003817

=10038171003817100381710038171003817119864 minus 119860

1119881421198611minus 1198621119882421198631minus 1198602119881421198612minus 1198622119882421198632

minus 1198603119881119867

421198613minus 1198623119882119867

421198633minus 1198604119881119879

421198614minus 1198624119882119879

421198634

10038171003817100381710038171003817

= 66115 times 10minus11

(35)

5 Conclusions

The above Figure 1 shows the convergence curve for theresidual function 119877(119896) In this paper an iterative algorithmconstructed to solve a complex matrix equation with conju-gate and transpose of two unknowns of the form 119860

11198811198611+

11986211198821198631+11986021198811198612+11986221198821198632+11986031198811198671198613+11986231198821198671198633+11986041198811198791198614+

11986241198821198791198634

= 119864 is presented We proved that the iterativealgorithms always converge to the solution for any initialmatrices We stated and proved some lemmas and theoremswhere the solutions are obtained The proposed methodis illustrated by numerical example where the obtainednumerical results show that our technique is very neat andefficient

References

[1] F Ding P X Liu and J Ding ldquoIterative solutions of thegeneralized Sylvester matrix equations by using the hierarchicalidentification principlerdquo Applied Mathematics and Computa-tion vol 197 no 1 pp 41ndash50 2008

[2] F Ding and T Chen ldquoGradient based iterative algorithmsfor solving a class of matrix equationsrdquo IEEE Transactions onAutomatic Control vol 50 no 8 pp 1216ndash1221 2005

[3] FDing andTChen ldquoHierarchical gradient-based identificationof multivariable discrete-time systemsrdquo Automatica vol 41 no2 pp 315ndash325 2005

[4] F Ding and T Chen ldquoIterative least-squares solutions of cou-pled Sylvester matrix equationsrdquo Systems and Control Lettersvol 54 no 2 pp 95ndash107 2005

[5] F Ding and T Chen ldquoOn iterative solutions of general coupledmatrix equationsrdquo SIAM Journal on Control and Optimizationvol 44 no 6 pp 2269ndash2284 2006

[6] F Ding and T Chen ldquoHierarchical least squares identificationmethods for multivariable systemsrdquo IEEE Transactions onAutomatic Control vol 50 no 3 pp 397ndash402 2005

[7] F Ding and T Chen ldquoPerformance analysis ofmulti-innovationgradient type identification methodsrdquo Automatica vol 43 no1 pp 1ndash14 2007

[8] AGWuX ZengG RDuan andW JWu ldquoIterative solutionsto the extended Sylvester-conjugate matrix equationsrdquo AppliedMathematics and Computation vol 217 no 1 pp 130ndash142 2010

[9] B Zhou G R Duan and Z Y Li ldquoGradient based iterativealgorithm for solving coupled matrix equationsrdquo Systems andControl Letters vol 58 no 5 pp 327ndash333 2009

[10] J H Bevis F J Hall and R E Hartwing ldquoConsimilarity andthe matrix equation119860119883minus119883119861 = 119862rdquo in Current Trends inMatrixTheory pp 51ndash64 North-Holland New York NY USA 1987

[11] R A Horn and C R Johnson Matrix Analysis CambridgeUniversity Press Cambridge UK 1990

[12] H Liping ldquoConsimilarity of quaternion matrices and complexmatricesrdquo Linear Algebra and Its Applications vol 331 no 1ndash3pp 21ndash30 2001

[13] T Jiang X Cheng and L Chen ldquoAn algebraic relation betweenconsimilarity and similarity of complex matrices and its appli-cationsrdquo Journal of Physics A vol 39 no 29 pp 9215ndash92222006

[14] J H Bevis F J Hall and R E Hartwig ldquoThe matrix equation119860119883 minus 119883119861 = 119862 and its special casesrdquo SIAM Journal on MatrixAnalysis and Applications vol 9 no 3 pp 348ndash359 1988

[15] A G Wu G R Duan and H H Yu ldquoOn solutions of thematrix equations 119883119865 minus 119860119883 = 119862 and 119883119865 minus 119860119883 = 119862rdquo AppliedMathematics and Computation vol 183 no 2 pp 932ndash9412006

[16] A Navarra P L Odell and D M Young ldquoRepresentationof the general common solution to the matrix equations11986011198831198611= 1198621 11986021198831198612= 1198622with applicationsrdquo Computers and

Mathematics with Applications vol 41 no 7-8 pp 929ndash9352001

[17] J W Van der Woude ldquoOn the existence of a common solutionX to the matrix equations 119860

119894119883119861119895= 119862 (i j) E Irdquo Linear Algebra

and Its Applications vol 375 no 1ndash3 pp 135ndash145 2003[18] P Bhimasankaram ldquoCommon solutions to the linear matrix

equations 119860119883 = 119862119883119861 = 119863 and EXF = Grdquo Sankhya Series Avol 38 pp 404ndash409 1976

[19] S Kumar Mitra ldquoThe matrix equations 119860119883 = 119862119883119861 = 119863rdquoLinear Algebra and Its Applications vol 59 pp 171ndash181 1984

[20] S K Mitra ldquoA pair of simultaneous linear matrix equations11986011198831198611

= 1198621 11986021198831198612

= 1198622and a matrix programming

problemrdquo Linear Algebra and Its Applications vol 131 pp 107ndash123 1990

[21] M A Ramadan M A Abdel Naby and A M E BayoumildquoOn the explicit solutions of forms of the Sylvester and theYakubovich matrix equationsrdquo Mathematical and ComputerModelling vol 50 no 9-10 pp 1400ndash1408 2009

[22] K W E Chu ldquoSingular value and generalized singular valuedecompositions and the solution of linear matrix equationsrdquoLinear Algebra and Its Applications vol 88-89 pp 83ndash98 1987

Journal of Discrete Mathematics 13

[23] Y X Yuan ldquoThe optimal solution of linear matrix equationby matrix decompositions Mathrdquo Numerica Sinica vol 24 pp165ndash176 2002

[24] A P Liao and Y Lei ldquoLeast-squares solution with the mini-mum-norm for the matrix equation (119860119883119861 119866119883119867) = (119862119863)rdquoComputers and Mathematics with Applications vol 50 no 3-4pp 539ndash549 2005

[25] Y H Liu ldquoRanks of least squares solutions of the matrix equa-tion 119860119883119861 = 119862rdquo Computers and Mathematics with Applicationsvol 55 no 6 pp 1270ndash1278 2008

[26] X Sheng and G Chen ldquoA finite iterative method for solving apair of linear matrix equations (119860119883119861 119862119883119863) = (119864 119865)rdquo AppliedMathematics and Computation vol 189 no 2 pp 1350ndash13582007

[27] A G Wu W Liu and G R Duan ldquoOn the conjugate productof complex polynomial matricesrdquo Mathematical and ComputerModelling vol 53 no 9-10 pp 2031ndash2043 2011

[28] A GWu G FengW Liu and G R Duan ldquoThe complete solu-tion to the Sylvester-polynomial-conjugate matrix equationsrdquoMathematical and Computer Modelling vol 53 no 9-10 pp2044ndash2056 2011

[29] A G Wu B Li Y Zhang and G R Duan ldquoFinite iterativesolutions to coupled Sylvester-conjugate matrix equationsrdquoApplied Mathematical Modelling vol 35 no 3 pp 1065ndash10802011

[30] AGWu L Lv andGRDuan ldquoIterative algorithms for solvinga class of complex conjugate and transpose matrix equationsrdquoApplied Mathematics and Computation vol 217 no 21 pp8343ndash8353 2011

[31] X ZhangMatrix Analysis and Application TsinghuaUniversityPress Beijing China 2004

[32] A G Wu L Lv and M Z Hou ldquoFinite iterative algorithms forextended Sylvester-conjugate matrix equationsrdquo Mathematicaland Computer Modelling vol 54 pp 2363ndash2384 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Discrete Mathematics 9

By using Algorithm 3 from (19) and induction we have

Re tr (119877119867119904+1

1198770)

= Re

tr[

[

(119877119904minus

10038171003817100381710038171198771199041003817100381710038171003817

2

10038171003817100381710038171198751199041003817100381710038171003817

2

+1003817100381710038171003817119876119904

1003817100381710038171003817

2

times (11986011198751199041198611+ 11986211198761199041198631+ 11986021198751199041198612

+ 11986221198761199041198632+ 1198603119875119867

1199041198613+ 1198623119876119867

1199041198633

+ 1198604119875119879

1199041198614+ 1198624119876119879

1199041198634))

119867

1198770]

]

= Re tr (1198771198671199041198770) minus

10038171003817100381710038171198771199041003817100381710038171003817

2

10038171003817100381710038171198751199041003817100381710038171003817

2

+1003817100381710038171003817119876119904

1003817100381710038171003817

2

times tr (119875119867119904119860119867

11198770119861119867

1+ 119876119867

119904119862119867

11198770119863119867

1

+ 119875119904

119867

119860119867

21198770119861119867

2+ 119876119904

119867

119862119867

21198770119863119867

2

+ 119875119904119860119867

31198770119861119867

3+ 119876119904119862119867

31198770119863119867

3

+ 119875119904119860119867

41198770119861119867

4+ 119876119904119862119867

41198770119863119867

4)

Re tr (119877119867119904+1

1198770)

= Re tr (1198771198671199041198770) minus

10038171003817100381710038171198771199041003817100381710038171003817

2

10038171003817100381710038171198751199041003817100381710038171003817

2

+1003817100381710038171003817119876119904

1003817100381710038171003817

2

times Re tr (119875119867119904119860119867

11198770119861119867

1+ 119876119867

119904119862119867

11198770119863119867

1

+ 119875119904

119867

119860119867

21198770119861119867

2+ 119876119904

119867

119862119867

21198770119863119867

2

+ 1198613119877119867

01198603119875119867

119904+ 1198633119877119867

01198623119876119867

119904

+ 1198614119877119879

01198604119875119867

119904+ 1198634119877119879

01198624119876119867

119904)

= minus

10038171003817100381710038171198771199041003817100381710038171003817

2

10038171003817100381710038171198751199041003817100381710038171003817

2

+1003817100381710038171003817119876119904

1003817100381710038171003817

2

times Re tr (119875119867119904

(119860119867

11198770119861119867

1+ 119860119867

21198770119861119867

2

+ 1198613119877119867

01198603+ 1198614119877119879

01198604)

+ 119876119867

119904(119862119867

11198770119863119867

1+ 119862119867

21198770119863119867

2

+1198633119877119867

01198623+ 1198634119877119879

01198624))

= minus

10038171003817100381710038171198771199041003817100381710038171003817

2

10038171003817100381710038171198751199041003817100381710038171003817

2

+1003817100381710038171003817119876119904

1003817100381710038171003817

2Re tr (119875119867

1199041198750+ 119876119867

1199041198760) = 0

Re tr (119875119867119904+1

1198750+ 119876119867

119904+11198760)

= Retr[(119860119867

1119877119904+1

119861119867

1+ 119860119867

2119877119904+1

119861119867

2

+ 1198613119877119867

119904+11198603+ 1198614119877119879

119904+11198604

+

1003817100381710038171003817119877119904+11003817100381710038171003817

2

10038171003817100381710038171198771199041003817100381710038171003817

2119875119904)

119867

1198750

+ (119862119867

1119877119904+1

119863119867

1+ 119862119867

2119877119904+1

119863119867

2

+ 1198633119877119867

119904+11198623+ 1198634119877119879

119896+11198624

+

1003817100381710038171003817119877119904+11003817100381710038171003817

2

10038171003817100381710038171198771199041003817100381710038171003817

2119876119904)

119867

1198760]

]

= Retr[119877119867119904+1

119860111987501198611+ 119877119904+1

119867

119860211987501198612

+ 119877119904+1

119861119867

31198750119860119867

3+ 119877119904+1

1198614

119867

11987501198604

119867

+

1003817100381710038171003817119877119904+11003817100381710038171003817

2

10038171003817100381710038171198771199041003817100381710038171003817

2119875119867

1199041198750+ 119877119867

119904+1119862111987601198631

+ 119877119904+1

119867

119862211987601198632+ 119877119904+1

119863119867

31198760119862119867

3

+ 119877119904+1

1198634

119867

11987601198624

119867

+

1003817100381710038171003817119877119904+11003817100381710038171003817

2

10038171003817100381710038171198771199041003817100381710038171003817

2119876119867

1199041198760]

= Retr[119877119867119904+1

119860111987501198611+ 119877119904+1

119867

119860211987501198612

+ 1198603119875119867

01198613119877119867

119904+1+ 1198604119875119879

01198614119877119867

119904+1

+

1003817100381710038171003817119877119904+11003817100381710038171003817

2

10038171003817100381710038171198771199041003817100381710038171003817

2119875119867

1199041198750+ 119877119867

119904+1119862111987601198631

+ 119877119904+1

119867

119862211987601198632+ 1198623119876119867

01198633119877119867

119904+1

+ 1198624119876119879

01198634119877119867

119904+1+

1003817100381710038171003817119877119904+11003817100381710038171003817

2

10038171003817100381710038171198771199041003817100381710038171003817

2119876119867

1199041198760]

= Retr[119877119867119904+1

(119860111987501198611+ 119862111987601198631+ 119860211987501198612

+ 119862211987601198632+ 1198603119875119867

01198613

+ 1198623119876119867

01198633+ 1198604119875119879

01198614

+ 1198624119876119879

01198634)

+

1003817100381710038171003817119877119904+11003817100381710038171003817

2

10038171003817100381710038171198771199041003817100381710038171003817

2(119875119867

1199041198750+ 119876119867

1199041198760)]

=

100381710038171003817100381711987501003817100381710038171003817

2

+10038171003817100381710038171198760

1003817100381710038171003817

2

100381710038171003817100381711987701003817100381710038171003817

2Re tr (119877119867

119904+1(1198770minus 1198771))= 0

(27)

Then (26) is holds

10 Journal of Discrete Mathematics

From Algorithm 3 we have

Re tr (119875119867119894+119904+1

119875119894+ 119876119867

119894+119904+1119876119894)

= Retr[(119860119867

1119877119894+119904+1

119861119867

1+ 119860119867

2119877119894+119904+1

119861119867

2

+ 1198613119877119867

119894+119904+11198603+ 1198614119877119879

119894+119904+11198604

+

1003817100381710038171003817119877119894+119904+11003817100381710038171003817

2

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2119875119894+119904

)

119867

119875119894

+ (119862119867

1119877119894+119904+1

119863119867

1+ 119862119867

2119877119894+119904+1

119863119867

2

+ 1198633119877119867

119894+119904+11198623+ 1198634119877119879

119894+119904+11198624

+

1003817100381710038171003817119877119894+119904+11003817100381710038171003817

2

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2119876119894+119904

)

119867

119876119894]

]

= Retr[119877119867119894+119904+1

11986011198751198941198611+ 119877119894+119904+1

119867

11986021198751198941198612

+ 119877119894+119904+1

119861119867

3119875119894119860119867

3+ 119877119894+119904+1

1198614

119867

1198751198941198604

119867

+

1003817100381710038171003817119877119894+119904+11003817100381710038171003817

2

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2119875119867

119894+119904119875119894+ 119877119867

119894+119904+111986211198761198941198631

+ 119877119894+119904+1

119867

11986221198761198941198632+ 119877119894+119904+1

119863119867

3119876119894119862119867

3

+ 119877119894+119904+1

1198634

119867

1198761198941198624

119867

+

1003817100381710038171003817119877119894+119904+11003817100381710038171003817

2

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2119876119867

119894+119904119876119894]

= Retr[119877119867119894+119904+1

11986011198751198941198611+ 119877119894+119904+1

119867

11986021198751198941198612

+ 1198603119875119867

1198941198613119877119867

119894+119904+1+ 1198604119875119879

1198941198614119877119867

119894+119904+1

+

1003817100381710038171003817119877119894+119904+11003817100381710038171003817

2

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2119875119867

119894+119904119875119894+ 119877119867

119894+119904+111986211198761198941198631

+ 119877119894+119904+1

119867

11986221198761198941198632+ 1198623119876119867

1198941198633119877119867

119894+119904+1

+ 1198624119876119879

1198961198634119877119867

119894+119904+1+

1003817100381710038171003817119877119894+119904+11003817100381710038171003817

2

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2119876119867

119894+119904119876119894]

= Retr[119877119867119894+119904+1

(11986011198751198941198611+ 11986211198761198941198631

+ 11986021198751198941198612+ 11986221198761198941198632

+ 1198603119875119867

1198941198613+ 1198623119875119867

1198941198633

+ 1198604119875119879

1198941198614+ 1198624119876119879

1198941198634)

+

1003817100381710038171003817119877119894+119904+11003817100381710038171003817

2

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2(119875119867

119894+119904119875119894+ 119876119867

119894+119904119876119894)]

=

10038171003817100381710038171198751198941003817100381710038171003817

2

+1003817100381710038171003817119876119894

1003817100381710038171003817

2

10038171003817100381710038171198771198941003817100381710038171003817

2Re tr (119877119867

119894+119904+1(119877119894minus 119877119894+1

))

=

10038171003817100381710038171198751198941003817100381710038171003817

2

+1003817100381710038171003817119876119894

1003817100381710038171003817

2

10038171003817100381710038171198771198941003817100381710038171003817

2Re tr (119877119867

119894+119904+1119877119894)

(28)

Also from (19) we have

Re tr (119877119867119894+119904+1

119877119894)

= Re

tr[

[

(119877119894+119904

minus

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2

1003817100381710038171003817119875119894+1199041003817100381710038171003817

2

+1003817100381710038171003817119876119894+119904

1003817100381710038171003817

2

times (1198601119875119894+119904

1198611+ 1198621119876119894+119904

1198631

+ 1198602119875119894+119904

1198612+ 1198622119876119894+119904

1198632

+ 1198603119875119867

119894+1199041198613+ 1198623119876119867

119894+1199041198633

+ 1198604119875119879

119894+1199041198614+ 1198624119876119879

119894+1199041198634))

119867

119877119894]

]

= Retr (119877119867119894+119904

119877119894) minus

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2

1003817100381710038171003817119875119894+1199041003817100381710038171003817

2

+1003817100381710038171003817119876119894+119904

1003817100381710038171003817

2

times tr (119875119867119894+119904

119860119867

1119877119894119861119867

1+ 119876119867

119894+119904119862119867

1119877119894119863119867

1

+ 119875119894+119904

119867

119860119867

2119877119894119861119867

2+ 119876119894+119904

119867

119862119867

2119877119894119863119867

2

+ 119875119894+119904

119860119867

3119877119894119861119867

3+ 119876119894+119904

119862119867

3119877119894119863119867

3

+ 119875119894+119904

119860119867

4119877119894119861119867

4+ 119876119894+119904

119862119867

4119877119894119863119867

4)

Re tr (119877119867119894+119904+1

119877119894)

= minus

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2

1003817100381710038171003817119875119894+1199041003817100381710038171003817

2

+1003817100381710038171003817119876119894+119904

1003817100381710038171003817

2

times Retr(119875119867119894+119904

119860119867

1119877119894119861119867

1+ 119876119867

119894+119904119862119867

1119877119894119863119867

1

+ 119875119894+119904

119867

119860119867

2119877119894119861119867

2+ 119876119894+119904

119867

119862119867

2119877119894119863119867

2

+ 1198613119877119867

1198941198603119875119867

119894+119904+ 1198633119877119867

1198941198623119876119867

119894+119904

+ 1198614119877119879

1198941198604119875119867

119894+119904+ 1198634119877119879

1198941198624119876119867

119894+119904)

Journal of Discrete Mathematics 11

= minus

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2

1003817100381710038171003817119875119894+1199041003817100381710038171003817

2

+1003817100381710038171003817119876119894+119904

1003817100381710038171003817

2

times Re tr [119875119867119894+119904

(119860119867

1119877119894119861119867

1+ 119860119867

2119877119894119861119867

2

+ 1198613119877119867

1198941198603+ 1198614119877119879

1198941198604)

+ 119876119867

119894+119904(119862119867

1119877119894119863119867

1+ 119862119867

2119877119894119863119867

2

+ 1198633119877119867

1198941198623+ 1198634119877119879

1198941198624)]

= minus

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2

1003817100381710038171003817119875119894+1199041003817100381710038171003817

2

+1003817100381710038171003817119876119894+119904

1003817100381710038171003817

2

times Re tr[119875119867119894+119904

(119875119894minus

10038171003817100381710038171198771198941003817100381710038171003817

2

1003817100381710038171003817119877119894minus11003817100381710038171003817

2119875119894minus1

)

+ 119876119867

119894+119904(119876119894minus

10038171003817100381710038171198771198941003817100381710038171003817

2

1003817100381710038171003817119877119894minus11003817100381710038171003817

2119876119894minus1

)]

=

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2

1003817100381710038171003817119875119894+1199041003817100381710038171003817

2

+1003817100381710038171003817119876119894+119904

1003817100381710038171003817

2

10038171003817100381710038171198771198941003817100381710038171003817

2

1003817100381710038171003817119877119894minus11003817100381710038171003817

2

times Re tr (119875119867119894+119904

119875119894minus1

+ 119876119867

119894+119904119876119894minus1

)

(29)

Repeating (28) and (29) one can easily obtain for certain 120572

and 120573

tr (119875119867119894+119904+1

119875119894+ 119876119867

119894+119904+1119876119894) = 120572 [tr (119875119867

119904+11198751+ 119876119867

119904+11198761)]

tr (119877119867119894+119904+1

119877119894) = 120573 [tr (119877119867

119904+11198771)]

(30)

Combining these two relations with (26) implies that (24)holds for 119897 = 119904 + 1 From Steps 1 and 2 the conclusion holdsby the principle of inductionWith the above two lemmas wehave the following theorem

Theorem 6 (see [32]) If the matrix equation (1) is consistentthen a solution can be obtained within finite iteration steps byusing Algorithm 3 for any initial matrices 119881

11198821

4 Numerical Example

In this section we present numerical example to illustrate theapplication of our proposed methods

Example 7 In this example we illustrate our theoreticalresults of Algorithm 3 for solving the system of matrixequation

11986011198811198611+ 11986211198821198631+ 11986021198811198612+ 11986221198821198632+ 11986031198811198671198613

+ 11986231198821198671198633+ 11986041198811198791198614+ 11986241198821198791198634= 119864

(31)

Because of the influence of the error of calculation theresidual 119877(119896) is usually unequal to zero in this process ofthe iteration We regard the matrix 119877(119896) as a zero matrix if119877(119896) lt 10

minus10Given

1198601= [

2 + 3119894 minus119894 1 + 119894

5 1 + 2119894 minus3]

1198602= [

2 + 3119894 minus119894 1 + 119894

5 1 + 2119894 minus3]

1198603= [

0 2 minus 119894 119894

minus1 + 3119894 2 0] 119860

4=[

0 1 minus 3119894 1 + 119894

0 4 + 119894 minus3119894]

1198621= [

1 + 2119894 3 minus 119894 4

minus119894 2119894 minus3] 119862

2=[

3 + 2119894 0 1 + 119894

0 4119894 1 minus 2119894]

1198623= [

1 minus 3119894 2119894 minus3119894

1 2 + 3119894 4119894] 119862

4=[

1 minus 2119894 0 2

3 minus 119894 1 + 119894 minus1]

1198611= [

[

4 + 119894 minus119894

0 1 minus 119894

4119894 2 + 2119894

]

]

1198612= [

[

0 119894

1 + 119894 0

minus1 minus 119894 3119894

]

]

1198613= [

[

0 1

minus3119894 4 + 119894

5 1 + 2119894

]

]

1198614=[

[

3 + 119894 minus1 minus 119894

0 2 minus 119894

minus1 + 119894 2

]

]

1198631= [

[

0 0

1 minus 3119894 minus119894

2119894 minus3119894

]

]

1198632= [

[

0 119894

1 + 119894 0

minus1 minus 119894 3119894

]

]

1198633= [

[

0 1

minus3119894 4 + 119894

5 1 + 2119894

]

]

1198634= [

[

3119894 minus2 + 119894

0 119894

minus2119894 minus4119894

]

]

119864 = [42 + 55119894 115 + 25119894

minus38 minus 119894 132 + 44119894]

(32)

Taking 1198811= [0 0

0 0

0 0

] and 1198821= [0 0

0 0] we apply Algorithm 3

to compute 119881119896119882119896

And iterating 42 steps we get

119881

=[

[

00126 + 18415119894 00827 + 06381119894 11221 minus 09428119894

minus06903 + 10185119894 18818 + 10203119894 09208 + 04569119894

05344 minus 04909119894 09280 + 07169119894 04872 minus 02734119894

]

]

119882

=[

[

04218 minus 09710119894 01763 + 05183119894 11331 minus 00432119894

06273 minus 01216119894 minus03902 + 04313119894 06240 + 09828119894

minus07011 minus 03418119894 03695 + 17627119894 minus03032 + 07073119894

]

]

(33)

which satisfy the matrix equation

11986011198811198611+ 11986211198821198631+ 11986021198811198612+ 11986221198821198632+ 11986031198811198671198613

+ 11986231198821198671198633+ 11986041198811198791198614+ 11986241198821198791198634= 119864

(34)

12 Journal of Discrete Mathematics

log10 rk

4

2

0

minus2

minus4

minus6

minus8

minus100 5 10 15 20 25 30 35 40 45

k (iteration number)

Figure 1The relation between the number of iterations and residualfor the example

With the corresponding residual100381710038171003817100381711987742

1003817100381710038171003817

=10038171003817100381710038171003817119864 minus 119860

1119881421198611minus 1198621119882421198631minus 1198602119881421198612minus 1198622119882421198632

minus 1198603119881119867

421198613minus 1198623119882119867

421198633minus 1198604119881119879

421198614minus 1198624119882119879

421198634

10038171003817100381710038171003817

= 66115 times 10minus11

(35)

5 Conclusions

The above Figure 1 shows the convergence curve for theresidual function 119877(119896) In this paper an iterative algorithmconstructed to solve a complex matrix equation with conju-gate and transpose of two unknowns of the form 119860

11198811198611+

11986211198821198631+11986021198811198612+11986221198821198632+11986031198811198671198613+11986231198821198671198633+11986041198811198791198614+

11986241198821198791198634

= 119864 is presented We proved that the iterativealgorithms always converge to the solution for any initialmatrices We stated and proved some lemmas and theoremswhere the solutions are obtained The proposed methodis illustrated by numerical example where the obtainednumerical results show that our technique is very neat andefficient

References

[1] F Ding P X Liu and J Ding ldquoIterative solutions of thegeneralized Sylvester matrix equations by using the hierarchicalidentification principlerdquo Applied Mathematics and Computa-tion vol 197 no 1 pp 41ndash50 2008

[2] F Ding and T Chen ldquoGradient based iterative algorithmsfor solving a class of matrix equationsrdquo IEEE Transactions onAutomatic Control vol 50 no 8 pp 1216ndash1221 2005

[3] FDing andTChen ldquoHierarchical gradient-based identificationof multivariable discrete-time systemsrdquo Automatica vol 41 no2 pp 315ndash325 2005

[4] F Ding and T Chen ldquoIterative least-squares solutions of cou-pled Sylvester matrix equationsrdquo Systems and Control Lettersvol 54 no 2 pp 95ndash107 2005

[5] F Ding and T Chen ldquoOn iterative solutions of general coupledmatrix equationsrdquo SIAM Journal on Control and Optimizationvol 44 no 6 pp 2269ndash2284 2006

[6] F Ding and T Chen ldquoHierarchical least squares identificationmethods for multivariable systemsrdquo IEEE Transactions onAutomatic Control vol 50 no 3 pp 397ndash402 2005

[7] F Ding and T Chen ldquoPerformance analysis ofmulti-innovationgradient type identification methodsrdquo Automatica vol 43 no1 pp 1ndash14 2007

[8] AGWuX ZengG RDuan andW JWu ldquoIterative solutionsto the extended Sylvester-conjugate matrix equationsrdquo AppliedMathematics and Computation vol 217 no 1 pp 130ndash142 2010

[9] B Zhou G R Duan and Z Y Li ldquoGradient based iterativealgorithm for solving coupled matrix equationsrdquo Systems andControl Letters vol 58 no 5 pp 327ndash333 2009

[10] J H Bevis F J Hall and R E Hartwing ldquoConsimilarity andthe matrix equation119860119883minus119883119861 = 119862rdquo in Current Trends inMatrixTheory pp 51ndash64 North-Holland New York NY USA 1987

[11] R A Horn and C R Johnson Matrix Analysis CambridgeUniversity Press Cambridge UK 1990

[12] H Liping ldquoConsimilarity of quaternion matrices and complexmatricesrdquo Linear Algebra and Its Applications vol 331 no 1ndash3pp 21ndash30 2001

[13] T Jiang X Cheng and L Chen ldquoAn algebraic relation betweenconsimilarity and similarity of complex matrices and its appli-cationsrdquo Journal of Physics A vol 39 no 29 pp 9215ndash92222006

[14] J H Bevis F J Hall and R E Hartwig ldquoThe matrix equation119860119883 minus 119883119861 = 119862 and its special casesrdquo SIAM Journal on MatrixAnalysis and Applications vol 9 no 3 pp 348ndash359 1988

[15] A G Wu G R Duan and H H Yu ldquoOn solutions of thematrix equations 119883119865 minus 119860119883 = 119862 and 119883119865 minus 119860119883 = 119862rdquo AppliedMathematics and Computation vol 183 no 2 pp 932ndash9412006

[16] A Navarra P L Odell and D M Young ldquoRepresentationof the general common solution to the matrix equations11986011198831198611= 1198621 11986021198831198612= 1198622with applicationsrdquo Computers and

Mathematics with Applications vol 41 no 7-8 pp 929ndash9352001

[17] J W Van der Woude ldquoOn the existence of a common solutionX to the matrix equations 119860

119894119883119861119895= 119862 (i j) E Irdquo Linear Algebra

and Its Applications vol 375 no 1ndash3 pp 135ndash145 2003[18] P Bhimasankaram ldquoCommon solutions to the linear matrix

equations 119860119883 = 119862119883119861 = 119863 and EXF = Grdquo Sankhya Series Avol 38 pp 404ndash409 1976

[19] S Kumar Mitra ldquoThe matrix equations 119860119883 = 119862119883119861 = 119863rdquoLinear Algebra and Its Applications vol 59 pp 171ndash181 1984

[20] S K Mitra ldquoA pair of simultaneous linear matrix equations11986011198831198611

= 1198621 11986021198831198612

= 1198622and a matrix programming

problemrdquo Linear Algebra and Its Applications vol 131 pp 107ndash123 1990

[21] M A Ramadan M A Abdel Naby and A M E BayoumildquoOn the explicit solutions of forms of the Sylvester and theYakubovich matrix equationsrdquo Mathematical and ComputerModelling vol 50 no 9-10 pp 1400ndash1408 2009

[22] K W E Chu ldquoSingular value and generalized singular valuedecompositions and the solution of linear matrix equationsrdquoLinear Algebra and Its Applications vol 88-89 pp 83ndash98 1987

Journal of Discrete Mathematics 13

[23] Y X Yuan ldquoThe optimal solution of linear matrix equationby matrix decompositions Mathrdquo Numerica Sinica vol 24 pp165ndash176 2002

[24] A P Liao and Y Lei ldquoLeast-squares solution with the mini-mum-norm for the matrix equation (119860119883119861 119866119883119867) = (119862119863)rdquoComputers and Mathematics with Applications vol 50 no 3-4pp 539ndash549 2005

[25] Y H Liu ldquoRanks of least squares solutions of the matrix equa-tion 119860119883119861 = 119862rdquo Computers and Mathematics with Applicationsvol 55 no 6 pp 1270ndash1278 2008

[26] X Sheng and G Chen ldquoA finite iterative method for solving apair of linear matrix equations (119860119883119861 119862119883119863) = (119864 119865)rdquo AppliedMathematics and Computation vol 189 no 2 pp 1350ndash13582007

[27] A G Wu W Liu and G R Duan ldquoOn the conjugate productof complex polynomial matricesrdquo Mathematical and ComputerModelling vol 53 no 9-10 pp 2031ndash2043 2011

[28] A GWu G FengW Liu and G R Duan ldquoThe complete solu-tion to the Sylvester-polynomial-conjugate matrix equationsrdquoMathematical and Computer Modelling vol 53 no 9-10 pp2044ndash2056 2011

[29] A G Wu B Li Y Zhang and G R Duan ldquoFinite iterativesolutions to coupled Sylvester-conjugate matrix equationsrdquoApplied Mathematical Modelling vol 35 no 3 pp 1065ndash10802011

[30] AGWu L Lv andGRDuan ldquoIterative algorithms for solvinga class of complex conjugate and transpose matrix equationsrdquoApplied Mathematics and Computation vol 217 no 21 pp8343ndash8353 2011

[31] X ZhangMatrix Analysis and Application TsinghuaUniversityPress Beijing China 2004

[32] A G Wu L Lv and M Z Hou ldquoFinite iterative algorithms forextended Sylvester-conjugate matrix equationsrdquo Mathematicaland Computer Modelling vol 54 pp 2363ndash2384 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Journal of Discrete Mathematics

From Algorithm 3 we have

Re tr (119875119867119894+119904+1

119875119894+ 119876119867

119894+119904+1119876119894)

= Retr[(119860119867

1119877119894+119904+1

119861119867

1+ 119860119867

2119877119894+119904+1

119861119867

2

+ 1198613119877119867

119894+119904+11198603+ 1198614119877119879

119894+119904+11198604

+

1003817100381710038171003817119877119894+119904+11003817100381710038171003817

2

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2119875119894+119904

)

119867

119875119894

+ (119862119867

1119877119894+119904+1

119863119867

1+ 119862119867

2119877119894+119904+1

119863119867

2

+ 1198633119877119867

119894+119904+11198623+ 1198634119877119879

119894+119904+11198624

+

1003817100381710038171003817119877119894+119904+11003817100381710038171003817

2

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2119876119894+119904

)

119867

119876119894]

]

= Retr[119877119867119894+119904+1

11986011198751198941198611+ 119877119894+119904+1

119867

11986021198751198941198612

+ 119877119894+119904+1

119861119867

3119875119894119860119867

3+ 119877119894+119904+1

1198614

119867

1198751198941198604

119867

+

1003817100381710038171003817119877119894+119904+11003817100381710038171003817

2

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2119875119867

119894+119904119875119894+ 119877119867

119894+119904+111986211198761198941198631

+ 119877119894+119904+1

119867

11986221198761198941198632+ 119877119894+119904+1

119863119867

3119876119894119862119867

3

+ 119877119894+119904+1

1198634

119867

1198761198941198624

119867

+

1003817100381710038171003817119877119894+119904+11003817100381710038171003817

2

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2119876119867

119894+119904119876119894]

= Retr[119877119867119894+119904+1

11986011198751198941198611+ 119877119894+119904+1

119867

11986021198751198941198612

+ 1198603119875119867

1198941198613119877119867

119894+119904+1+ 1198604119875119879

1198941198614119877119867

119894+119904+1

+

1003817100381710038171003817119877119894+119904+11003817100381710038171003817

2

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2119875119867

119894+119904119875119894+ 119877119867

119894+119904+111986211198761198941198631

+ 119877119894+119904+1

119867

11986221198761198941198632+ 1198623119876119867

1198941198633119877119867

119894+119904+1

+ 1198624119876119879

1198961198634119877119867

119894+119904+1+

1003817100381710038171003817119877119894+119904+11003817100381710038171003817

2

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2119876119867

119894+119904119876119894]

= Retr[119877119867119894+119904+1

(11986011198751198941198611+ 11986211198761198941198631

+ 11986021198751198941198612+ 11986221198761198941198632

+ 1198603119875119867

1198941198613+ 1198623119875119867

1198941198633

+ 1198604119875119879

1198941198614+ 1198624119876119879

1198941198634)

+

1003817100381710038171003817119877119894+119904+11003817100381710038171003817

2

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2(119875119867

119894+119904119875119894+ 119876119867

119894+119904119876119894)]

=

10038171003817100381710038171198751198941003817100381710038171003817

2

+1003817100381710038171003817119876119894

1003817100381710038171003817

2

10038171003817100381710038171198771198941003817100381710038171003817

2Re tr (119877119867

119894+119904+1(119877119894minus 119877119894+1

))

=

10038171003817100381710038171198751198941003817100381710038171003817

2

+1003817100381710038171003817119876119894

1003817100381710038171003817

2

10038171003817100381710038171198771198941003817100381710038171003817

2Re tr (119877119867

119894+119904+1119877119894)

(28)

Also from (19) we have

Re tr (119877119867119894+119904+1

119877119894)

= Re

tr[

[

(119877119894+119904

minus

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2

1003817100381710038171003817119875119894+1199041003817100381710038171003817

2

+1003817100381710038171003817119876119894+119904

1003817100381710038171003817

2

times (1198601119875119894+119904

1198611+ 1198621119876119894+119904

1198631

+ 1198602119875119894+119904

1198612+ 1198622119876119894+119904

1198632

+ 1198603119875119867

119894+1199041198613+ 1198623119876119867

119894+1199041198633

+ 1198604119875119879

119894+1199041198614+ 1198624119876119879

119894+1199041198634))

119867

119877119894]

]

= Retr (119877119867119894+119904

119877119894) minus

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2

1003817100381710038171003817119875119894+1199041003817100381710038171003817

2

+1003817100381710038171003817119876119894+119904

1003817100381710038171003817

2

times tr (119875119867119894+119904

119860119867

1119877119894119861119867

1+ 119876119867

119894+119904119862119867

1119877119894119863119867

1

+ 119875119894+119904

119867

119860119867

2119877119894119861119867

2+ 119876119894+119904

119867

119862119867

2119877119894119863119867

2

+ 119875119894+119904

119860119867

3119877119894119861119867

3+ 119876119894+119904

119862119867

3119877119894119863119867

3

+ 119875119894+119904

119860119867

4119877119894119861119867

4+ 119876119894+119904

119862119867

4119877119894119863119867

4)

Re tr (119877119867119894+119904+1

119877119894)

= minus

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2

1003817100381710038171003817119875119894+1199041003817100381710038171003817

2

+1003817100381710038171003817119876119894+119904

1003817100381710038171003817

2

times Retr(119875119867119894+119904

119860119867

1119877119894119861119867

1+ 119876119867

119894+119904119862119867

1119877119894119863119867

1

+ 119875119894+119904

119867

119860119867

2119877119894119861119867

2+ 119876119894+119904

119867

119862119867

2119877119894119863119867

2

+ 1198613119877119867

1198941198603119875119867

119894+119904+ 1198633119877119867

1198941198623119876119867

119894+119904

+ 1198614119877119879

1198941198604119875119867

119894+119904+ 1198634119877119879

1198941198624119876119867

119894+119904)

Journal of Discrete Mathematics 11

= minus

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2

1003817100381710038171003817119875119894+1199041003817100381710038171003817

2

+1003817100381710038171003817119876119894+119904

1003817100381710038171003817

2

times Re tr [119875119867119894+119904

(119860119867

1119877119894119861119867

1+ 119860119867

2119877119894119861119867

2

+ 1198613119877119867

1198941198603+ 1198614119877119879

1198941198604)

+ 119876119867

119894+119904(119862119867

1119877119894119863119867

1+ 119862119867

2119877119894119863119867

2

+ 1198633119877119867

1198941198623+ 1198634119877119879

1198941198624)]

= minus

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2

1003817100381710038171003817119875119894+1199041003817100381710038171003817

2

+1003817100381710038171003817119876119894+119904

1003817100381710038171003817

2

times Re tr[119875119867119894+119904

(119875119894minus

10038171003817100381710038171198771198941003817100381710038171003817

2

1003817100381710038171003817119877119894minus11003817100381710038171003817

2119875119894minus1

)

+ 119876119867

119894+119904(119876119894minus

10038171003817100381710038171198771198941003817100381710038171003817

2

1003817100381710038171003817119877119894minus11003817100381710038171003817

2119876119894minus1

)]

=

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2

1003817100381710038171003817119875119894+1199041003817100381710038171003817

2

+1003817100381710038171003817119876119894+119904

1003817100381710038171003817

2

10038171003817100381710038171198771198941003817100381710038171003817

2

1003817100381710038171003817119877119894minus11003817100381710038171003817

2

times Re tr (119875119867119894+119904

119875119894minus1

+ 119876119867

119894+119904119876119894minus1

)

(29)

Repeating (28) and (29) one can easily obtain for certain 120572

and 120573

tr (119875119867119894+119904+1

119875119894+ 119876119867

119894+119904+1119876119894) = 120572 [tr (119875119867

119904+11198751+ 119876119867

119904+11198761)]

tr (119877119867119894+119904+1

119877119894) = 120573 [tr (119877119867

119904+11198771)]

(30)

Combining these two relations with (26) implies that (24)holds for 119897 = 119904 + 1 From Steps 1 and 2 the conclusion holdsby the principle of inductionWith the above two lemmas wehave the following theorem

Theorem 6 (see [32]) If the matrix equation (1) is consistentthen a solution can be obtained within finite iteration steps byusing Algorithm 3 for any initial matrices 119881

11198821

4 Numerical Example

In this section we present numerical example to illustrate theapplication of our proposed methods

Example 7 In this example we illustrate our theoreticalresults of Algorithm 3 for solving the system of matrixequation

11986011198811198611+ 11986211198821198631+ 11986021198811198612+ 11986221198821198632+ 11986031198811198671198613

+ 11986231198821198671198633+ 11986041198811198791198614+ 11986241198821198791198634= 119864

(31)

Because of the influence of the error of calculation theresidual 119877(119896) is usually unequal to zero in this process ofthe iteration We regard the matrix 119877(119896) as a zero matrix if119877(119896) lt 10

minus10Given

1198601= [

2 + 3119894 minus119894 1 + 119894

5 1 + 2119894 minus3]

1198602= [

2 + 3119894 minus119894 1 + 119894

5 1 + 2119894 minus3]

1198603= [

0 2 minus 119894 119894

minus1 + 3119894 2 0] 119860

4=[

0 1 minus 3119894 1 + 119894

0 4 + 119894 minus3119894]

1198621= [

1 + 2119894 3 minus 119894 4

minus119894 2119894 minus3] 119862

2=[

3 + 2119894 0 1 + 119894

0 4119894 1 minus 2119894]

1198623= [

1 minus 3119894 2119894 minus3119894

1 2 + 3119894 4119894] 119862

4=[

1 minus 2119894 0 2

3 minus 119894 1 + 119894 minus1]

1198611= [

[

4 + 119894 minus119894

0 1 minus 119894

4119894 2 + 2119894

]

]

1198612= [

[

0 119894

1 + 119894 0

minus1 minus 119894 3119894

]

]

1198613= [

[

0 1

minus3119894 4 + 119894

5 1 + 2119894

]

]

1198614=[

[

3 + 119894 minus1 minus 119894

0 2 minus 119894

minus1 + 119894 2

]

]

1198631= [

[

0 0

1 minus 3119894 minus119894

2119894 minus3119894

]

]

1198632= [

[

0 119894

1 + 119894 0

minus1 minus 119894 3119894

]

]

1198633= [

[

0 1

minus3119894 4 + 119894

5 1 + 2119894

]

]

1198634= [

[

3119894 minus2 + 119894

0 119894

minus2119894 minus4119894

]

]

119864 = [42 + 55119894 115 + 25119894

minus38 minus 119894 132 + 44119894]

(32)

Taking 1198811= [0 0

0 0

0 0

] and 1198821= [0 0

0 0] we apply Algorithm 3

to compute 119881119896119882119896

And iterating 42 steps we get

119881

=[

[

00126 + 18415119894 00827 + 06381119894 11221 minus 09428119894

minus06903 + 10185119894 18818 + 10203119894 09208 + 04569119894

05344 minus 04909119894 09280 + 07169119894 04872 minus 02734119894

]

]

119882

=[

[

04218 minus 09710119894 01763 + 05183119894 11331 minus 00432119894

06273 minus 01216119894 minus03902 + 04313119894 06240 + 09828119894

minus07011 minus 03418119894 03695 + 17627119894 minus03032 + 07073119894

]

]

(33)

which satisfy the matrix equation

11986011198811198611+ 11986211198821198631+ 11986021198811198612+ 11986221198821198632+ 11986031198811198671198613

+ 11986231198821198671198633+ 11986041198811198791198614+ 11986241198821198791198634= 119864

(34)

12 Journal of Discrete Mathematics

log10 rk

4

2

0

minus2

minus4

minus6

minus8

minus100 5 10 15 20 25 30 35 40 45

k (iteration number)

Figure 1The relation between the number of iterations and residualfor the example

With the corresponding residual100381710038171003817100381711987742

1003817100381710038171003817

=10038171003817100381710038171003817119864 minus 119860

1119881421198611minus 1198621119882421198631minus 1198602119881421198612minus 1198622119882421198632

minus 1198603119881119867

421198613minus 1198623119882119867

421198633minus 1198604119881119879

421198614minus 1198624119882119879

421198634

10038171003817100381710038171003817

= 66115 times 10minus11

(35)

5 Conclusions

The above Figure 1 shows the convergence curve for theresidual function 119877(119896) In this paper an iterative algorithmconstructed to solve a complex matrix equation with conju-gate and transpose of two unknowns of the form 119860

11198811198611+

11986211198821198631+11986021198811198612+11986221198821198632+11986031198811198671198613+11986231198821198671198633+11986041198811198791198614+

11986241198821198791198634

= 119864 is presented We proved that the iterativealgorithms always converge to the solution for any initialmatrices We stated and proved some lemmas and theoremswhere the solutions are obtained The proposed methodis illustrated by numerical example where the obtainednumerical results show that our technique is very neat andefficient

References

[1] F Ding P X Liu and J Ding ldquoIterative solutions of thegeneralized Sylvester matrix equations by using the hierarchicalidentification principlerdquo Applied Mathematics and Computa-tion vol 197 no 1 pp 41ndash50 2008

[2] F Ding and T Chen ldquoGradient based iterative algorithmsfor solving a class of matrix equationsrdquo IEEE Transactions onAutomatic Control vol 50 no 8 pp 1216ndash1221 2005

[3] FDing andTChen ldquoHierarchical gradient-based identificationof multivariable discrete-time systemsrdquo Automatica vol 41 no2 pp 315ndash325 2005

[4] F Ding and T Chen ldquoIterative least-squares solutions of cou-pled Sylvester matrix equationsrdquo Systems and Control Lettersvol 54 no 2 pp 95ndash107 2005

[5] F Ding and T Chen ldquoOn iterative solutions of general coupledmatrix equationsrdquo SIAM Journal on Control and Optimizationvol 44 no 6 pp 2269ndash2284 2006

[6] F Ding and T Chen ldquoHierarchical least squares identificationmethods for multivariable systemsrdquo IEEE Transactions onAutomatic Control vol 50 no 3 pp 397ndash402 2005

[7] F Ding and T Chen ldquoPerformance analysis ofmulti-innovationgradient type identification methodsrdquo Automatica vol 43 no1 pp 1ndash14 2007

[8] AGWuX ZengG RDuan andW JWu ldquoIterative solutionsto the extended Sylvester-conjugate matrix equationsrdquo AppliedMathematics and Computation vol 217 no 1 pp 130ndash142 2010

[9] B Zhou G R Duan and Z Y Li ldquoGradient based iterativealgorithm for solving coupled matrix equationsrdquo Systems andControl Letters vol 58 no 5 pp 327ndash333 2009

[10] J H Bevis F J Hall and R E Hartwing ldquoConsimilarity andthe matrix equation119860119883minus119883119861 = 119862rdquo in Current Trends inMatrixTheory pp 51ndash64 North-Holland New York NY USA 1987

[11] R A Horn and C R Johnson Matrix Analysis CambridgeUniversity Press Cambridge UK 1990

[12] H Liping ldquoConsimilarity of quaternion matrices and complexmatricesrdquo Linear Algebra and Its Applications vol 331 no 1ndash3pp 21ndash30 2001

[13] T Jiang X Cheng and L Chen ldquoAn algebraic relation betweenconsimilarity and similarity of complex matrices and its appli-cationsrdquo Journal of Physics A vol 39 no 29 pp 9215ndash92222006

[14] J H Bevis F J Hall and R E Hartwig ldquoThe matrix equation119860119883 minus 119883119861 = 119862 and its special casesrdquo SIAM Journal on MatrixAnalysis and Applications vol 9 no 3 pp 348ndash359 1988

[15] A G Wu G R Duan and H H Yu ldquoOn solutions of thematrix equations 119883119865 minus 119860119883 = 119862 and 119883119865 minus 119860119883 = 119862rdquo AppliedMathematics and Computation vol 183 no 2 pp 932ndash9412006

[16] A Navarra P L Odell and D M Young ldquoRepresentationof the general common solution to the matrix equations11986011198831198611= 1198621 11986021198831198612= 1198622with applicationsrdquo Computers and

Mathematics with Applications vol 41 no 7-8 pp 929ndash9352001

[17] J W Van der Woude ldquoOn the existence of a common solutionX to the matrix equations 119860

119894119883119861119895= 119862 (i j) E Irdquo Linear Algebra

and Its Applications vol 375 no 1ndash3 pp 135ndash145 2003[18] P Bhimasankaram ldquoCommon solutions to the linear matrix

equations 119860119883 = 119862119883119861 = 119863 and EXF = Grdquo Sankhya Series Avol 38 pp 404ndash409 1976

[19] S Kumar Mitra ldquoThe matrix equations 119860119883 = 119862119883119861 = 119863rdquoLinear Algebra and Its Applications vol 59 pp 171ndash181 1984

[20] S K Mitra ldquoA pair of simultaneous linear matrix equations11986011198831198611

= 1198621 11986021198831198612

= 1198622and a matrix programming

problemrdquo Linear Algebra and Its Applications vol 131 pp 107ndash123 1990

[21] M A Ramadan M A Abdel Naby and A M E BayoumildquoOn the explicit solutions of forms of the Sylvester and theYakubovich matrix equationsrdquo Mathematical and ComputerModelling vol 50 no 9-10 pp 1400ndash1408 2009

[22] K W E Chu ldquoSingular value and generalized singular valuedecompositions and the solution of linear matrix equationsrdquoLinear Algebra and Its Applications vol 88-89 pp 83ndash98 1987

Journal of Discrete Mathematics 13

[23] Y X Yuan ldquoThe optimal solution of linear matrix equationby matrix decompositions Mathrdquo Numerica Sinica vol 24 pp165ndash176 2002

[24] A P Liao and Y Lei ldquoLeast-squares solution with the mini-mum-norm for the matrix equation (119860119883119861 119866119883119867) = (119862119863)rdquoComputers and Mathematics with Applications vol 50 no 3-4pp 539ndash549 2005

[25] Y H Liu ldquoRanks of least squares solutions of the matrix equa-tion 119860119883119861 = 119862rdquo Computers and Mathematics with Applicationsvol 55 no 6 pp 1270ndash1278 2008

[26] X Sheng and G Chen ldquoA finite iterative method for solving apair of linear matrix equations (119860119883119861 119862119883119863) = (119864 119865)rdquo AppliedMathematics and Computation vol 189 no 2 pp 1350ndash13582007

[27] A G Wu W Liu and G R Duan ldquoOn the conjugate productof complex polynomial matricesrdquo Mathematical and ComputerModelling vol 53 no 9-10 pp 2031ndash2043 2011

[28] A GWu G FengW Liu and G R Duan ldquoThe complete solu-tion to the Sylvester-polynomial-conjugate matrix equationsrdquoMathematical and Computer Modelling vol 53 no 9-10 pp2044ndash2056 2011

[29] A G Wu B Li Y Zhang and G R Duan ldquoFinite iterativesolutions to coupled Sylvester-conjugate matrix equationsrdquoApplied Mathematical Modelling vol 35 no 3 pp 1065ndash10802011

[30] AGWu L Lv andGRDuan ldquoIterative algorithms for solvinga class of complex conjugate and transpose matrix equationsrdquoApplied Mathematics and Computation vol 217 no 21 pp8343ndash8353 2011

[31] X ZhangMatrix Analysis and Application TsinghuaUniversityPress Beijing China 2004

[32] A G Wu L Lv and M Z Hou ldquoFinite iterative algorithms forextended Sylvester-conjugate matrix equationsrdquo Mathematicaland Computer Modelling vol 54 pp 2363ndash2384 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Discrete Mathematics 11

= minus

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2

1003817100381710038171003817119875119894+1199041003817100381710038171003817

2

+1003817100381710038171003817119876119894+119904

1003817100381710038171003817

2

times Re tr [119875119867119894+119904

(119860119867

1119877119894119861119867

1+ 119860119867

2119877119894119861119867

2

+ 1198613119877119867

1198941198603+ 1198614119877119879

1198941198604)

+ 119876119867

119894+119904(119862119867

1119877119894119863119867

1+ 119862119867

2119877119894119863119867

2

+ 1198633119877119867

1198941198623+ 1198634119877119879

1198941198624)]

= minus

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2

1003817100381710038171003817119875119894+1199041003817100381710038171003817

2

+1003817100381710038171003817119876119894+119904

1003817100381710038171003817

2

times Re tr[119875119867119894+119904

(119875119894minus

10038171003817100381710038171198771198941003817100381710038171003817

2

1003817100381710038171003817119877119894minus11003817100381710038171003817

2119875119894minus1

)

+ 119876119867

119894+119904(119876119894minus

10038171003817100381710038171198771198941003817100381710038171003817

2

1003817100381710038171003817119877119894minus11003817100381710038171003817

2119876119894minus1

)]

=

1003817100381710038171003817119877119894+1199041003817100381710038171003817

2

1003817100381710038171003817119875119894+1199041003817100381710038171003817

2

+1003817100381710038171003817119876119894+119904

1003817100381710038171003817

2

10038171003817100381710038171198771198941003817100381710038171003817

2

1003817100381710038171003817119877119894minus11003817100381710038171003817

2

times Re tr (119875119867119894+119904

119875119894minus1

+ 119876119867

119894+119904119876119894minus1

)

(29)

Repeating (28) and (29) one can easily obtain for certain 120572

and 120573

tr (119875119867119894+119904+1

119875119894+ 119876119867

119894+119904+1119876119894) = 120572 [tr (119875119867

119904+11198751+ 119876119867

119904+11198761)]

tr (119877119867119894+119904+1

119877119894) = 120573 [tr (119877119867

119904+11198771)]

(30)

Combining these two relations with (26) implies that (24)holds for 119897 = 119904 + 1 From Steps 1 and 2 the conclusion holdsby the principle of inductionWith the above two lemmas wehave the following theorem

Theorem 6 (see [32]) If the matrix equation (1) is consistentthen a solution can be obtained within finite iteration steps byusing Algorithm 3 for any initial matrices 119881

11198821

4 Numerical Example

In this section we present numerical example to illustrate theapplication of our proposed methods

Example 7 In this example we illustrate our theoreticalresults of Algorithm 3 for solving the system of matrixequation

11986011198811198611+ 11986211198821198631+ 11986021198811198612+ 11986221198821198632+ 11986031198811198671198613

+ 11986231198821198671198633+ 11986041198811198791198614+ 11986241198821198791198634= 119864

(31)

Because of the influence of the error of calculation theresidual 119877(119896) is usually unequal to zero in this process ofthe iteration We regard the matrix 119877(119896) as a zero matrix if119877(119896) lt 10

minus10Given

1198601= [

2 + 3119894 minus119894 1 + 119894

5 1 + 2119894 minus3]

1198602= [

2 + 3119894 minus119894 1 + 119894

5 1 + 2119894 minus3]

1198603= [

0 2 minus 119894 119894

minus1 + 3119894 2 0] 119860

4=[

0 1 minus 3119894 1 + 119894

0 4 + 119894 minus3119894]

1198621= [

1 + 2119894 3 minus 119894 4

minus119894 2119894 minus3] 119862

2=[

3 + 2119894 0 1 + 119894

0 4119894 1 minus 2119894]

1198623= [

1 minus 3119894 2119894 minus3119894

1 2 + 3119894 4119894] 119862

4=[

1 minus 2119894 0 2

3 minus 119894 1 + 119894 minus1]

1198611= [

[

4 + 119894 minus119894

0 1 minus 119894

4119894 2 + 2119894

]

]

1198612= [

[

0 119894

1 + 119894 0

minus1 minus 119894 3119894

]

]

1198613= [

[

0 1

minus3119894 4 + 119894

5 1 + 2119894

]

]

1198614=[

[

3 + 119894 minus1 minus 119894

0 2 minus 119894

minus1 + 119894 2

]

]

1198631= [

[

0 0

1 minus 3119894 minus119894

2119894 minus3119894

]

]

1198632= [

[

0 119894

1 + 119894 0

minus1 minus 119894 3119894

]

]

1198633= [

[

0 1

minus3119894 4 + 119894

5 1 + 2119894

]

]

1198634= [

[

3119894 minus2 + 119894

0 119894

minus2119894 minus4119894

]

]

119864 = [42 + 55119894 115 + 25119894

minus38 minus 119894 132 + 44119894]

(32)

Taking 1198811= [0 0

0 0

0 0

] and 1198821= [0 0

0 0] we apply Algorithm 3

to compute 119881119896119882119896

And iterating 42 steps we get

119881

=[

[

00126 + 18415119894 00827 + 06381119894 11221 minus 09428119894

minus06903 + 10185119894 18818 + 10203119894 09208 + 04569119894

05344 minus 04909119894 09280 + 07169119894 04872 minus 02734119894

]

]

119882

=[

[

04218 minus 09710119894 01763 + 05183119894 11331 minus 00432119894

06273 minus 01216119894 minus03902 + 04313119894 06240 + 09828119894

minus07011 minus 03418119894 03695 + 17627119894 minus03032 + 07073119894

]

]

(33)

which satisfy the matrix equation

11986011198811198611+ 11986211198821198631+ 11986021198811198612+ 11986221198821198632+ 11986031198811198671198613

+ 11986231198821198671198633+ 11986041198811198791198614+ 11986241198821198791198634= 119864

(34)

12 Journal of Discrete Mathematics

log10 rk

4

2

0

minus2

minus4

minus6

minus8

minus100 5 10 15 20 25 30 35 40 45

k (iteration number)

Figure 1The relation between the number of iterations and residualfor the example

With the corresponding residual100381710038171003817100381711987742

1003817100381710038171003817

=10038171003817100381710038171003817119864 minus 119860

1119881421198611minus 1198621119882421198631minus 1198602119881421198612minus 1198622119882421198632

minus 1198603119881119867

421198613minus 1198623119882119867

421198633minus 1198604119881119879

421198614minus 1198624119882119879

421198634

10038171003817100381710038171003817

= 66115 times 10minus11

(35)

5 Conclusions

The above Figure 1 shows the convergence curve for theresidual function 119877(119896) In this paper an iterative algorithmconstructed to solve a complex matrix equation with conju-gate and transpose of two unknowns of the form 119860

11198811198611+

11986211198821198631+11986021198811198612+11986221198821198632+11986031198811198671198613+11986231198821198671198633+11986041198811198791198614+

11986241198821198791198634

= 119864 is presented We proved that the iterativealgorithms always converge to the solution for any initialmatrices We stated and proved some lemmas and theoremswhere the solutions are obtained The proposed methodis illustrated by numerical example where the obtainednumerical results show that our technique is very neat andefficient

References

[1] F Ding P X Liu and J Ding ldquoIterative solutions of thegeneralized Sylvester matrix equations by using the hierarchicalidentification principlerdquo Applied Mathematics and Computa-tion vol 197 no 1 pp 41ndash50 2008

[2] F Ding and T Chen ldquoGradient based iterative algorithmsfor solving a class of matrix equationsrdquo IEEE Transactions onAutomatic Control vol 50 no 8 pp 1216ndash1221 2005

[3] FDing andTChen ldquoHierarchical gradient-based identificationof multivariable discrete-time systemsrdquo Automatica vol 41 no2 pp 315ndash325 2005

[4] F Ding and T Chen ldquoIterative least-squares solutions of cou-pled Sylvester matrix equationsrdquo Systems and Control Lettersvol 54 no 2 pp 95ndash107 2005

[5] F Ding and T Chen ldquoOn iterative solutions of general coupledmatrix equationsrdquo SIAM Journal on Control and Optimizationvol 44 no 6 pp 2269ndash2284 2006

[6] F Ding and T Chen ldquoHierarchical least squares identificationmethods for multivariable systemsrdquo IEEE Transactions onAutomatic Control vol 50 no 3 pp 397ndash402 2005

[7] F Ding and T Chen ldquoPerformance analysis ofmulti-innovationgradient type identification methodsrdquo Automatica vol 43 no1 pp 1ndash14 2007

[8] AGWuX ZengG RDuan andW JWu ldquoIterative solutionsto the extended Sylvester-conjugate matrix equationsrdquo AppliedMathematics and Computation vol 217 no 1 pp 130ndash142 2010

[9] B Zhou G R Duan and Z Y Li ldquoGradient based iterativealgorithm for solving coupled matrix equationsrdquo Systems andControl Letters vol 58 no 5 pp 327ndash333 2009

[10] J H Bevis F J Hall and R E Hartwing ldquoConsimilarity andthe matrix equation119860119883minus119883119861 = 119862rdquo in Current Trends inMatrixTheory pp 51ndash64 North-Holland New York NY USA 1987

[11] R A Horn and C R Johnson Matrix Analysis CambridgeUniversity Press Cambridge UK 1990

[12] H Liping ldquoConsimilarity of quaternion matrices and complexmatricesrdquo Linear Algebra and Its Applications vol 331 no 1ndash3pp 21ndash30 2001

[13] T Jiang X Cheng and L Chen ldquoAn algebraic relation betweenconsimilarity and similarity of complex matrices and its appli-cationsrdquo Journal of Physics A vol 39 no 29 pp 9215ndash92222006

[14] J H Bevis F J Hall and R E Hartwig ldquoThe matrix equation119860119883 minus 119883119861 = 119862 and its special casesrdquo SIAM Journal on MatrixAnalysis and Applications vol 9 no 3 pp 348ndash359 1988

[15] A G Wu G R Duan and H H Yu ldquoOn solutions of thematrix equations 119883119865 minus 119860119883 = 119862 and 119883119865 minus 119860119883 = 119862rdquo AppliedMathematics and Computation vol 183 no 2 pp 932ndash9412006

[16] A Navarra P L Odell and D M Young ldquoRepresentationof the general common solution to the matrix equations11986011198831198611= 1198621 11986021198831198612= 1198622with applicationsrdquo Computers and

Mathematics with Applications vol 41 no 7-8 pp 929ndash9352001

[17] J W Van der Woude ldquoOn the existence of a common solutionX to the matrix equations 119860

119894119883119861119895= 119862 (i j) E Irdquo Linear Algebra

and Its Applications vol 375 no 1ndash3 pp 135ndash145 2003[18] P Bhimasankaram ldquoCommon solutions to the linear matrix

equations 119860119883 = 119862119883119861 = 119863 and EXF = Grdquo Sankhya Series Avol 38 pp 404ndash409 1976

[19] S Kumar Mitra ldquoThe matrix equations 119860119883 = 119862119883119861 = 119863rdquoLinear Algebra and Its Applications vol 59 pp 171ndash181 1984

[20] S K Mitra ldquoA pair of simultaneous linear matrix equations11986011198831198611

= 1198621 11986021198831198612

= 1198622and a matrix programming

problemrdquo Linear Algebra and Its Applications vol 131 pp 107ndash123 1990

[21] M A Ramadan M A Abdel Naby and A M E BayoumildquoOn the explicit solutions of forms of the Sylvester and theYakubovich matrix equationsrdquo Mathematical and ComputerModelling vol 50 no 9-10 pp 1400ndash1408 2009

[22] K W E Chu ldquoSingular value and generalized singular valuedecompositions and the solution of linear matrix equationsrdquoLinear Algebra and Its Applications vol 88-89 pp 83ndash98 1987

Journal of Discrete Mathematics 13

[23] Y X Yuan ldquoThe optimal solution of linear matrix equationby matrix decompositions Mathrdquo Numerica Sinica vol 24 pp165ndash176 2002

[24] A P Liao and Y Lei ldquoLeast-squares solution with the mini-mum-norm for the matrix equation (119860119883119861 119866119883119867) = (119862119863)rdquoComputers and Mathematics with Applications vol 50 no 3-4pp 539ndash549 2005

[25] Y H Liu ldquoRanks of least squares solutions of the matrix equa-tion 119860119883119861 = 119862rdquo Computers and Mathematics with Applicationsvol 55 no 6 pp 1270ndash1278 2008

[26] X Sheng and G Chen ldquoA finite iterative method for solving apair of linear matrix equations (119860119883119861 119862119883119863) = (119864 119865)rdquo AppliedMathematics and Computation vol 189 no 2 pp 1350ndash13582007

[27] A G Wu W Liu and G R Duan ldquoOn the conjugate productof complex polynomial matricesrdquo Mathematical and ComputerModelling vol 53 no 9-10 pp 2031ndash2043 2011

[28] A GWu G FengW Liu and G R Duan ldquoThe complete solu-tion to the Sylvester-polynomial-conjugate matrix equationsrdquoMathematical and Computer Modelling vol 53 no 9-10 pp2044ndash2056 2011

[29] A G Wu B Li Y Zhang and G R Duan ldquoFinite iterativesolutions to coupled Sylvester-conjugate matrix equationsrdquoApplied Mathematical Modelling vol 35 no 3 pp 1065ndash10802011

[30] AGWu L Lv andGRDuan ldquoIterative algorithms for solvinga class of complex conjugate and transpose matrix equationsrdquoApplied Mathematics and Computation vol 217 no 21 pp8343ndash8353 2011

[31] X ZhangMatrix Analysis and Application TsinghuaUniversityPress Beijing China 2004

[32] A G Wu L Lv and M Z Hou ldquoFinite iterative algorithms forextended Sylvester-conjugate matrix equationsrdquo Mathematicaland Computer Modelling vol 54 pp 2363ndash2384 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Journal of Discrete Mathematics

log10 rk

4

2

0

minus2

minus4

minus6

minus8

minus100 5 10 15 20 25 30 35 40 45

k (iteration number)

Figure 1The relation between the number of iterations and residualfor the example

With the corresponding residual100381710038171003817100381711987742

1003817100381710038171003817

=10038171003817100381710038171003817119864 minus 119860

1119881421198611minus 1198621119882421198631minus 1198602119881421198612minus 1198622119882421198632

minus 1198603119881119867

421198613minus 1198623119882119867

421198633minus 1198604119881119879

421198614minus 1198624119882119879

421198634

10038171003817100381710038171003817

= 66115 times 10minus11

(35)

5 Conclusions

The above Figure 1 shows the convergence curve for theresidual function 119877(119896) In this paper an iterative algorithmconstructed to solve a complex matrix equation with conju-gate and transpose of two unknowns of the form 119860

11198811198611+

11986211198821198631+11986021198811198612+11986221198821198632+11986031198811198671198613+11986231198821198671198633+11986041198811198791198614+

11986241198821198791198634

= 119864 is presented We proved that the iterativealgorithms always converge to the solution for any initialmatrices We stated and proved some lemmas and theoremswhere the solutions are obtained The proposed methodis illustrated by numerical example where the obtainednumerical results show that our technique is very neat andefficient

References

[1] F Ding P X Liu and J Ding ldquoIterative solutions of thegeneralized Sylvester matrix equations by using the hierarchicalidentification principlerdquo Applied Mathematics and Computa-tion vol 197 no 1 pp 41ndash50 2008

[2] F Ding and T Chen ldquoGradient based iterative algorithmsfor solving a class of matrix equationsrdquo IEEE Transactions onAutomatic Control vol 50 no 8 pp 1216ndash1221 2005

[3] FDing andTChen ldquoHierarchical gradient-based identificationof multivariable discrete-time systemsrdquo Automatica vol 41 no2 pp 315ndash325 2005

[4] F Ding and T Chen ldquoIterative least-squares solutions of cou-pled Sylvester matrix equationsrdquo Systems and Control Lettersvol 54 no 2 pp 95ndash107 2005

[5] F Ding and T Chen ldquoOn iterative solutions of general coupledmatrix equationsrdquo SIAM Journal on Control and Optimizationvol 44 no 6 pp 2269ndash2284 2006

[6] F Ding and T Chen ldquoHierarchical least squares identificationmethods for multivariable systemsrdquo IEEE Transactions onAutomatic Control vol 50 no 3 pp 397ndash402 2005

[7] F Ding and T Chen ldquoPerformance analysis ofmulti-innovationgradient type identification methodsrdquo Automatica vol 43 no1 pp 1ndash14 2007

[8] AGWuX ZengG RDuan andW JWu ldquoIterative solutionsto the extended Sylvester-conjugate matrix equationsrdquo AppliedMathematics and Computation vol 217 no 1 pp 130ndash142 2010

[9] B Zhou G R Duan and Z Y Li ldquoGradient based iterativealgorithm for solving coupled matrix equationsrdquo Systems andControl Letters vol 58 no 5 pp 327ndash333 2009

[10] J H Bevis F J Hall and R E Hartwing ldquoConsimilarity andthe matrix equation119860119883minus119883119861 = 119862rdquo in Current Trends inMatrixTheory pp 51ndash64 North-Holland New York NY USA 1987

[11] R A Horn and C R Johnson Matrix Analysis CambridgeUniversity Press Cambridge UK 1990

[12] H Liping ldquoConsimilarity of quaternion matrices and complexmatricesrdquo Linear Algebra and Its Applications vol 331 no 1ndash3pp 21ndash30 2001

[13] T Jiang X Cheng and L Chen ldquoAn algebraic relation betweenconsimilarity and similarity of complex matrices and its appli-cationsrdquo Journal of Physics A vol 39 no 29 pp 9215ndash92222006

[14] J H Bevis F J Hall and R E Hartwig ldquoThe matrix equation119860119883 minus 119883119861 = 119862 and its special casesrdquo SIAM Journal on MatrixAnalysis and Applications vol 9 no 3 pp 348ndash359 1988

[15] A G Wu G R Duan and H H Yu ldquoOn solutions of thematrix equations 119883119865 minus 119860119883 = 119862 and 119883119865 minus 119860119883 = 119862rdquo AppliedMathematics and Computation vol 183 no 2 pp 932ndash9412006

[16] A Navarra P L Odell and D M Young ldquoRepresentationof the general common solution to the matrix equations11986011198831198611= 1198621 11986021198831198612= 1198622with applicationsrdquo Computers and

Mathematics with Applications vol 41 no 7-8 pp 929ndash9352001

[17] J W Van der Woude ldquoOn the existence of a common solutionX to the matrix equations 119860

119894119883119861119895= 119862 (i j) E Irdquo Linear Algebra

and Its Applications vol 375 no 1ndash3 pp 135ndash145 2003[18] P Bhimasankaram ldquoCommon solutions to the linear matrix

equations 119860119883 = 119862119883119861 = 119863 and EXF = Grdquo Sankhya Series Avol 38 pp 404ndash409 1976

[19] S Kumar Mitra ldquoThe matrix equations 119860119883 = 119862119883119861 = 119863rdquoLinear Algebra and Its Applications vol 59 pp 171ndash181 1984

[20] S K Mitra ldquoA pair of simultaneous linear matrix equations11986011198831198611

= 1198621 11986021198831198612

= 1198622and a matrix programming

problemrdquo Linear Algebra and Its Applications vol 131 pp 107ndash123 1990

[21] M A Ramadan M A Abdel Naby and A M E BayoumildquoOn the explicit solutions of forms of the Sylvester and theYakubovich matrix equationsrdquo Mathematical and ComputerModelling vol 50 no 9-10 pp 1400ndash1408 2009

[22] K W E Chu ldquoSingular value and generalized singular valuedecompositions and the solution of linear matrix equationsrdquoLinear Algebra and Its Applications vol 88-89 pp 83ndash98 1987

Journal of Discrete Mathematics 13

[23] Y X Yuan ldquoThe optimal solution of linear matrix equationby matrix decompositions Mathrdquo Numerica Sinica vol 24 pp165ndash176 2002

[24] A P Liao and Y Lei ldquoLeast-squares solution with the mini-mum-norm for the matrix equation (119860119883119861 119866119883119867) = (119862119863)rdquoComputers and Mathematics with Applications vol 50 no 3-4pp 539ndash549 2005

[25] Y H Liu ldquoRanks of least squares solutions of the matrix equa-tion 119860119883119861 = 119862rdquo Computers and Mathematics with Applicationsvol 55 no 6 pp 1270ndash1278 2008

[26] X Sheng and G Chen ldquoA finite iterative method for solving apair of linear matrix equations (119860119883119861 119862119883119863) = (119864 119865)rdquo AppliedMathematics and Computation vol 189 no 2 pp 1350ndash13582007

[27] A G Wu W Liu and G R Duan ldquoOn the conjugate productof complex polynomial matricesrdquo Mathematical and ComputerModelling vol 53 no 9-10 pp 2031ndash2043 2011

[28] A GWu G FengW Liu and G R Duan ldquoThe complete solu-tion to the Sylvester-polynomial-conjugate matrix equationsrdquoMathematical and Computer Modelling vol 53 no 9-10 pp2044ndash2056 2011

[29] A G Wu B Li Y Zhang and G R Duan ldquoFinite iterativesolutions to coupled Sylvester-conjugate matrix equationsrdquoApplied Mathematical Modelling vol 35 no 3 pp 1065ndash10802011

[30] AGWu L Lv andGRDuan ldquoIterative algorithms for solvinga class of complex conjugate and transpose matrix equationsrdquoApplied Mathematics and Computation vol 217 no 21 pp8343ndash8353 2011

[31] X ZhangMatrix Analysis and Application TsinghuaUniversityPress Beijing China 2004

[32] A G Wu L Lv and M Z Hou ldquoFinite iterative algorithms forextended Sylvester-conjugate matrix equationsrdquo Mathematicaland Computer Modelling vol 54 pp 2363ndash2384 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Discrete Mathematics 13

[23] Y X Yuan ldquoThe optimal solution of linear matrix equationby matrix decompositions Mathrdquo Numerica Sinica vol 24 pp165ndash176 2002

[24] A P Liao and Y Lei ldquoLeast-squares solution with the mini-mum-norm for the matrix equation (119860119883119861 119866119883119867) = (119862119863)rdquoComputers and Mathematics with Applications vol 50 no 3-4pp 539ndash549 2005

[25] Y H Liu ldquoRanks of least squares solutions of the matrix equa-tion 119860119883119861 = 119862rdquo Computers and Mathematics with Applicationsvol 55 no 6 pp 1270ndash1278 2008

[26] X Sheng and G Chen ldquoA finite iterative method for solving apair of linear matrix equations (119860119883119861 119862119883119863) = (119864 119865)rdquo AppliedMathematics and Computation vol 189 no 2 pp 1350ndash13582007

[27] A G Wu W Liu and G R Duan ldquoOn the conjugate productof complex polynomial matricesrdquo Mathematical and ComputerModelling vol 53 no 9-10 pp 2031ndash2043 2011

[28] A GWu G FengW Liu and G R Duan ldquoThe complete solu-tion to the Sylvester-polynomial-conjugate matrix equationsrdquoMathematical and Computer Modelling vol 53 no 9-10 pp2044ndash2056 2011

[29] A G Wu B Li Y Zhang and G R Duan ldquoFinite iterativesolutions to coupled Sylvester-conjugate matrix equationsrdquoApplied Mathematical Modelling vol 35 no 3 pp 1065ndash10802011

[30] AGWu L Lv andGRDuan ldquoIterative algorithms for solvinga class of complex conjugate and transpose matrix equationsrdquoApplied Mathematics and Computation vol 217 no 21 pp8343ndash8353 2011

[31] X ZhangMatrix Analysis and Application TsinghuaUniversityPress Beijing China 2004

[32] A G Wu L Lv and M Z Hou ldquoFinite iterative algorithms forextended Sylvester-conjugate matrix equationsrdquo Mathematicaland Computer Modelling vol 54 pp 2363ndash2384 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of