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Inequalites for monotonic pairs of Z-matricesRonald L. Smith a & Shu-An Hu ba Department of Mathematics , University of Tennessee at Chattanooga , Chattanooga, TN37403-2598b Department of Mathematics , Springfield College , Springfield, MA 01109-3797Published online: 31 Mar 2008.

To cite this article: Ronald L. Smith & Shu-An Hu (1998) Inequalites for monotonic pairs of Z-matrices, Linear andMultilinear Algebra, 44:1, 57-65, DOI: 10.1080/03081089808818548

To link to this article: http://dx.doi.org/10.1080/03081089808818548

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Inequalities for Monotonic Pairs of Z-Matrices

RONALD L. SMITHa.* and SHU-AN H U ~

aDepartrnent of Mathematics, University of Tennessee at Chattanooga, Chattanooga, TN 37403-2598; b~epartrnent of Mathematics, Springfield College, Springfield, MA 07709-3797

Communicated by S. Pierce

(Received 6 August 1996; In final form 7 April 1997)

Let A and B be n-by-n Z-matrices satisfying A 5 B entrywise. It is shown that. if A and B are in the class of Z-matrices consisting of No-matrices and .M-matrices. then A + B is Itself either an No-matrix or an M-matrix; inequalities for the detcrrninant and the inverse of A + B are obtained as well.

Kcnt,ords: .M-matrix: determinant; spectral radius; principal submatrix

AMS Subject Clas.tificcrrions: 1 5 A 4 , 15A45

1. INTRODUCTION

In general, certain classes of Z-matrices (for example. M-matrices and No-matrices) are closed under scalar multiplication and positive diagonal scaling, but are not closed under addition. For example. A = [ l o 121 and B = [!, ', 1 are both M-matrices but A + B is not.

However, in [2], Ky Fan showed that if A and B are nonsingular M-matrices satisfying A 5 B entrywise, then A -- B is also a non- singular M-matrix. Also, for certain Z-matrices A and B belonging to

*The research of t h ~ s author has supported, In part, bq the CECA Scholara Program of the Center of Excellence In Computer Applicat~ons at the Uni\ers~ty of Tennessce at Chattanooga

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58 R. L. SMITH A N D S.-A. HU

KO or No and satisfying A < B and certain restrictive semipositivity conditions, Y. S. Lee [5, 61 was able to show that A + B is either an M- matrix (if det(A + B) > 0) or an No-matrix (if det(A + B) < 0). Lee also established inequalities for the determinant and the inverse of A+ B. In this paper, we will show that Lee's semipositivity conditions on A and B are unnecessary and establish his results, as well as several related results, without them. We will also correct several inaccuracies in Lee's paper.

2. PRELIMINARIES AND NOTATION

Throughout this paper, we assume all matrices are n-by-n with real entries. A Z-matrix is an n-by-n real matrix whose off-diagonal entries are nonpositive. Those Z-matrices of the form

in which r > p(B), the spectral radius of B, are called (nonsingular) M- matrices, and, if t = p(B), singular M-matrices. Following Fiedler and Ptak [3] , we let Z denote the class of all Z-matrices, K the class of nonsingular M-matrices, and KO the class of general M-matrices, i.e., matrices of the form (1) in which t > p(B). Thus Ko\K denotes the class of singular M-matrices. M-matrices have various applications in diverse fields such as economics and linear programming where one is concerned with the convergence of iterative solutions and\or the existence of nonnegative solutions of systems of linear and nonlinear equations.

Ky Fan [2] defined N-matrices to be matrices of form (I) , with t

chosen so that p,,-, (B) < t < p(B). Here p,, I (B) denotes the maximum spectral radius of all (n-1)-by-(n-I) principal submatrices of B. G. Johnson [4] studied a class of matrices closely related to Ky Fan's N-matrices, i.e., he considered matrices of the form (1) in which p,,+,(B) < t < p(B) and called them No-matrices.

We shall need the following well known characterizations of K, KO, Kol,K, and No, respectively.

THEOREM 2.1 [ I , Theorem 6.2.3; 3. Theorem 4.31 Let AEZ. Then the .following are equivalent.

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Z-MATRICES

( i) A M . (ii) All principal minors of A are nonnegative and det A > 0.

(iii) All principal minors of A are positive. (iv) All leading prihcipal minors o f A are positive.

THEOREM 2.2 [3, Theorem 5.11 Let A E Z . Then the follo~.ing are equivalent.

( i) A€&. (ii) All principal minors of A are nonnegative.

(iii) A + SIEK for each 5 > 0.

Directly from Theorems 1.1 and 1.2, we have

THEOK EM 2.3 Let A E Z . Then the following are equivalent.

( i ) A€Ko:K. (ii) All proper prinicipal minors of A are nonnegative and det A = 0 .

(iii) A is singular and A + SIEK for each 6 > 0.

THEOREM 2.4 [4, Theorem 2.71 Let A E Z . Then the following are equivalent.

( i ) A € N o . (ii) All proper principal minors of A are nonnegative and det A < 0.

(iii) A-' exists, A-' < 0, and A-' is irreducible.

Following Berman and Plemmons [I], we write A 2 B if aii 2 b, for all i and j, A > B if A 2 B and A f B , and A>>B if aii > b, for all i and j.

For a square matrix A we let AT denote the transpose of A.

3. MAIN RESULTS

In this section we are mainly concerned with the following question: given two 2-matrices A , B lying in KO U No and satisfying A 5 B , are positive linear combinations of A and B themselves in either K, Ko\K, or No?. Since each of these Z-matrix classes is closed under positive scalar multiplication, we may restrict the proofs to the case in which we have a convex combination of A and B. Of course, analogous results for the sum A + B follow as corollaries.

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60 K . L. SMITH AND S.-A HIJ

A proof of our tirst lemma was originally given in [6] based upon the folloning atatemcnt [5, Lemma 7: 6, Lemma 61.

Statement 3.1 I x t ,4eZ. IT Ai, > 0 for some v >> 0 , then det A > 0. Furthermore.

( a ) del A 0 implies that A E K and (b) det A = O if and only if A Y = ~ . In this case, if A is irreducible, we

have A E K ~ .

We note that thc iniplication "if det A = 0, then Av = 0" is false as shown by the following example.

If c is the %vector consisting of all ones, Ae = [0. 0 , l lT# [O, 0 , Oll. Lee's proof was based upon 3.1 .b (although Lee claims to be using

3.1 .a,). Thus, the original proof of our first lemma is incorrect, and we now give a corrcct proof.

L E M M A 3.3 Suppose fhcit A E K ~ I\' und B E L r~?tli A < B, upzd let u , h 0. I f thcrc, c ~ i r t r ti>O ~ u c h thcit Au = 0 , thm CIA& hB E K.

Proof' Suppose that A€K,,\,K and BEK with A < B, that there exists u>>O such that .4u = 0, and that C - N A + ( I -rr)B, 0 < rr < I . Obviously CEZ, so we wi!l show that C E K by showing all principal minors of C' are positive.

Now Bu ;. 0 since H, A. Thus? O dct(uA) - det((l - u)B) < det C' by [S. Thcorom 3.101 and continuity. Let C l = u A i I + ( 1 u)Bll bc a k-by- k principal submatrix of C', 1 5 k n. Without loss of generality, we may assume

Partition u = [ u l . u2lT conformally with A . Then Au=O implies A l l u l > O , since A I 2 u 2 1 0 . Since B > A , we have B I 1 u l ~ A I i u l > O . Again by [S], sir~cc A I and B I 1 E K, we have det C1 1 2 det(aA, +

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Z-MATRICES 6 1

det((1 -n)BI > 0. Thus. all principal minors of C are positive, which implies C'EK. 0

(i) of the following lemma was noted by Ky Fan [2] and the last part was proved by Fiedler and Ptak [3, Theorem 4.61 We include the short proofs for completeness.

L E M M A 3.4 Lct A, B E K ~tsitli A 5 B. Then, f i r . (ill (I. b > 0.

PI oof Let A , B t K w ~ t h A < B m d let a. h < 0 Then ( u t b) A 5 ([A IIB 5 ( u + h)B and aA + ~ B E K bq [2. Lemma I ] (11) m d ( u ~ ) then follow from [3. Theorem 4 61

Since A t K , there exi\ts u>O. such that Aa>O Hence. Bu > Au >> O and (AB ' ) ~ u = AU>O Since A B ' < I, this ~mplies . ~ B - ' E K Aftel ~uhlng trdnsposes. we hdle an andogous proof for B ' A 0

The following theorem was proved 111 [6. Lemma 8 and Theorem 91 under the senl~positivity aas~~mption that there exist u. 1 >>0 such that n u = n T , =O.

T H E O R F M 3.5 Let A€&, und BEK tt.ith A < B. Tllcw, Jw (dl (1, h < 0 ,

Further.. ~f K , then AB-I, B ' A E K ~ K.

Pi oof I f 4 c K . u e are done bq Lemma 3 4 So 'tssume that A t K o K (so that 4 R), m d that E=crA+(l-a)B, ahe ie 0 < t i < 1 We ulll p iole the theorem by induction on n, the order of A and B If iz = 1. the theorem 1s obv~ously true So assume the theorem holds for dl1 such matr~ces of order < iz. A = 31-C and B = JI-D for somc C 2 D 0 and some 3 > 0 Smce AEK? K and BcK. J= p(C) > p(D) There exists zr 2 0 , uf 0. such that Cu = p(C)u If u>O, (I) holds by Lemmd 3 3 Otherwise, there exists an n-by-n pelmutation matrix P such that Pu = 1 = [::I, where v 1 >0 has h components and i - 0 Partition

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62 R. L. SMITH AND S.-A. HU

PAP^, P B P ~ , P C P ~ , P D P ~ and PEP= conformally with u so that, for instance, PCP= has the partitioned form

where C' l l is k x k . Since P C P ' V = ~ ( C ) V , we have C l I q =p(C)v l and Czl v l = 0. This, in turn, implies p(C1 = p(C) and C2' = 0. Since C 2 D 2 0 , Dzl = 0 also. Thus,

p,,T [El I El21 0 E22

Since A l = p(C)I-CI Mo\K and B I I EK, El E K by induction. Now if p(C2,) = p(C), then A2, = p(C)I-C2,€Ko\K and since B,,c K, E,,EK by induction. On the other hand, if p(C2,) < p(C3, then &,=p(C) I-C2,EK and thus &EK by Lemma 3.4. In either case, (i) is established. (ii) and (iii) follow from the proof of Lemma 3.4.

For f > 0, consider AB- ' t t I = ( A + FB)B I. A+ F B (by (i)) and (1 + c)B are both in K and A + t B 5 (1 + 6)B. Thus, ( A + t B ) [(I + E)B] ' = ( A 4 cB) (1 + e) ' B P ' E K by Lemma 3.4. This implies A B ' + ~ E K since K is closed under positive scalar multiplication. Since d e t ( ~ ~ - ' ) =o , it follows by continuity that AB-'E Ko\K. After taking transposes, an analogous argument holds for B ' A . 0

Next we show that the sum of two singular M-matrices A and B satisfying A < B is itself a singular M-matrix.

THEOREM 3.6 Let A, B€Ko\K wirk A I B. Then, aA + bB€Ko\,K, for all a, h > 0 .

Proof' Suppose E = aA + (1 -u)B, where 0 < a < 1 . Let P = max{b,,} > max{u,,). Then, A = PI-C and B= PI-D, where C 2 D > 0. Since A ,

BE Ko\K, 3 = p(C) = p(D). If 0 5 a < I , C > aC + ( 1 -u)U 2 D which implies p(C) 2 p(aC+ (1-a)D) 2 p(D). Thus, p(aC+ (1 -a)D) = B also and hence E= PI-(uC+ (I - ~ ) D ) E Ko\K. 0

The following theorem shows that the sum of an No-matrix A and a singular M-matrix B satisfying A I B is itself an No-matrix and was

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proved in [6, Theorem 111 under the semipositivity assumption that there exists u>>O such that Bu = 0.

THEOREM 3.7 Let A€No and B€Ko\K with A < B. Then, aA +bB€ No, for all a, b > 0.

Proof Suppose E = aA + (1 -a)B, where 0 < a < 1. Let P = max{h,,} > max{a,,). Then, A = PI-C and B= PI-D, where C > D 2 0. Note that C is irreducible since A E N ~ . Thus, P = p(D) < p(C) [I, Corollary 1.51. Also, pnpl(C) 5 3 < p(C) since CE No. Thus, for 0 < a < 1, D < a C + (1 - a)D < C which implies p(D) < p(uC + (1 -a)D) < p(C) (by the irreduci- bility of C ) and P,,-~(UC t (1-a)D) < p,-'(C). Combining these spectral inequalities, we have pn-l(aCA (1 -a)D) 5 P = p(D) < p(aC+ (I -a)D) which in turn implies that E = 31-(aC+ (1-a)D)€No. This completes the proof. 0

It is easy to construct 2-by-2 examples to show that Theorem 3.7 does not necessarily hold if B is a nonsingular M-matrix rather than a singular M-matrix.

Example 3.8 Consider the No-matrix

If we consider the nonsingular M-matrices

B1 = [I, ;'I, B2 = [A y] and B3 = [i i], respectively, we see that A + B, EN,,, A + B2€K0\K, and A + B3€K.

The next theorem clarifies this situation and was proved in [6, Theorem 12 and Theorem 131 by assuming the following semipositivity condition: for all i, 1 < i < n, there exists an (n- 1)-vector v(i)>>O such that A(i) v(i) > 0, where A(i) denotes the principal submatrix obtained from A by deleting the ith row and ith column.

THEOREM 3.9 Let A€No and BEK with A < B. Then, for all a, h > 0,

(i) aA + BE K (f and only if det(aA + bB) > 0, (ii) aA + bBcKo\ K if and only if det(aA + bB) = 0, and

(iii) aA + bB€No if and only if det(aA + bB) < 0. Further, i f 0 < a 5 1 and C = aA + (1-a)B, then

(iv) detC > 0 implies a A 1 + ( I -u)B-' 5 C-', and (v) detC < 0 implies U A - I + ( I -U)B-' 2 C-'.

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64 R L. SMITH A N D S - A H U

Proof' By Lenimas 3.4, 3.5 and 3.6, all proper principal subniatrices of uA + hR are in KO. Then, (i) follows from [I. C8, p. 1351. (ii) follows from [I. A8. p.1491, and (iii) follows from [4. Lern~ua 2.11. The proof-s of (iv) and ( v ) are identical to those given in [6. Theorem 131 under the afore-nientioncd semipositivity conditionb, but are included for conipleteness. First observe that u A - ~ ' t (1-cr)B ' -c-~ = a ( l -u) ( A - ' B+ K ' A - ~ I ) c ~ ' . Since A - I B. B - ' A 5 1 [6. Lemma 101. (iv) and (v) follow from (i) and (iii), respectively. 0

We state the following theorem [S. theorcm 141 and supply a different proof since the original proof of ( i ) given in [ 5 ] is not correct. For in that proof. it was incorrectly stated that by Corollary 3.1 1 of [8]. "all proper principal submatrices of t l A + f2B are 121-matrices (possibly singular)". However, for M-matrices A and B, the quoted corollary requires that there exists u>>O such that ,411 > 0 and Bti > 0; this obviously may not be the case. We also note that the inequality in part (ii) was incorrectly reversed in both thc statement and proof of the original result.

( I ) [ / A - hBc.Yo. (I,) dct (ir 4) - ciet(hB) 5 det(aA 4 bB). mil

(111) h ' R I < (l,A + 6 ~ ) ~ ' .

Proof ' We will just prove (i) and use the proof given in [S] for (ii) and (iii) with the correction cited in the remark preceding the statement of the theorem, It suficcs to prove (i) for E:= n i l + (I -a)B, where 0 < u < 1 . Let 3 - max{hil) 2 max jn,,: . Then, since A, Bc~%'~ , and A < B, A = 31-Cand B= 31--D, for some irreducible matrices C. D satisfying C > D > 0. Thus, pi, .l(D) 5 p i ,+ l (C)<3 p(D)

p ( C ' ) . For 0 < rr < I . we have p , , _ , ( ~ < - I (1 -n )D) p ,,... 1 ( C ) < 3 < p(D) < p(uC'+ (I -0)D) . Thisimplies E= 31- ( a C C ( 1 -u)D)€No.

Rtw~crrk It was shown in [7, Theorem 2.31 that, if A. B&VO ~vi th A 5 B. then A 'B, B A 'are nonsingular M-matrices (and, equiva- lently. F 'A. AB-' are inverse M-matrices. i.e.. tnatrices whose inverses are .M-matrices). However, a~nbiguities exist when A E N ~ and RE& with A 5 B as seen by the follo\ving example.

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Example 3.12 Consider the No-matrix A = [L3 -:] and the non- s i n g u l a r M - m a t r i x B = [tl -:] s a t i s f y i n g A 5 B. T h e n A-'B = 8 [ I -5 -:] is an No-matrix.

On the other hand, suppose we have the No-matrix C = [L2 -:] and the nonsingular M-matrix D = [ y , so that C I D. Then C-'D =

f [I: I:] which is not an No-matrix (in fact, it is the inverse of an No- matrix).

Acknowledgement

The authors wish to thank Professor William Watkins and the referee for their diligence in the review of this manuscript.

References

Berman, A. and Plemmons, R. (1979). Non-negative matrices in the mathematical sciences, Academic Press, New York. Ky. Fan (1964). Inequalities for the sum of two M-matrices, Proc. Koninkl. Ned. Akarl. Wetenschrrp. S m , A67, 602-610. Fiedler, M. and Ptak, V. (1962). On matrices with non-positive off diagonal elements and positive principal minors, Czech. Math. J., 12, 382-400. Johnson, G. (1982). A generalization of A-matrices. Linear Algebrcr and its Applicarions, 48. 201 - 21 7. Lee, Y. S. (1990). On the sum of two No-matrices, Linear and Multilinear Algebra, 26, 215-221. Lee, Y. S. (1991). Inequalities for M and No-matrices, Linear and Multilinear Algebra. 29, 149 - 154. Smith, R. (1986). On the spectrum of No-matrices, Linear Algebra and its Appliccrtions. 83. 129- 134. Smith. R. (1987). On Markham's M-matrix properties, Linear Algebra and its Applicurions, 87, 189- 195.

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