1966 - Potter - Matrix Quadratic Solutions

Matrix Quadratic SolutionsAuthor(s): James E. PotterSource: SIAM Journal on Applied Mathematics, Vol. 14, No. 3 (May, 1966), pp. 496-501Published by: Society for Industrial and Applied MathematicsStable URL: http://www.jstor.org/stable/2946224 .Accessed: 06/08/2014 20:37

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

Society for Industrial and Applied Mathematics is collaborating with JSTOR to digitize, preserve and extendaccess to SIAM Journal on Applied Mathematics.

http://www.jstor.org

This content downloaded from 200.130.19.173 on Wed, 6 Aug 2014 20:37:49 PMAll use subject to JSTOR Terms and Conditions

J. SIAM APPL. MATH. Vol. 14, No. 3, May, 1966

Printed in U.S.A.

MATRIX QUADRATIC SOLUTIONS*

JAMES E. POTTERt

Coinsider the matrix quadratic equation

(1) O = A + BX + XB*-XCX, in which the coefficients A, B, and C are given n by n matrices and the solution X is also required to be an n by n matrix. A formula expressing the solutions of (1) in terms of eigenvectors of the 2n by 2n partitioned matrix

M[C -B*]

formed from the coefficieint matrices in (1) will be obtained below. To sim- plify the theorems, it will be assumed that M has a diagonal Jordan canonical form.

It is necessary to solve (1) in order to find the steady state solutions of matrix Riccati differential equations [1], [2] with constant coefficients which arise in the theory of multiwire transmission lines [3], linear filtering and prediction [4], and optimum automatic control theory [5]. In the Wiener theory of filtering and prediction of stationary stochastic processes [6], the spectrum factorization step may be replaced by requiring instead the positive definite solution of a matrix quadratic equation. As will be seen below, this involves finding the eigenvectors of M. However, oiily the eigenvectors corresponding to eigenvalues with positive real parts are actually used in the solution. Thus, as might be expected, the solution of the matrix quadratic equation in this case is operationally similar to spectrum factorization.

In the following analysis capital letters will be used to denote square or rectangular matrices, lower case letters will denote column vectors, and lower case Greek letters will denote scalars. An asterisk will be used to represent the complex conjugate transpose of a matrix or column vector or the complex conjugate of a scalar. It will be necessary to have a notation for the upper and lower halves of eigenvectors of M. Thus if a is an eigenvector of M, we will write

b

* Received by the editors December 3, 1964, and in revised form April 30, 1965. t Experimental Astronomy Laboratory, Massachusetts Institute of Technology,

Cambridge, Massachusetts. This work was supported by the National Aeronautics and Space Administration under Grant NsG 254-62.

496


MATRIX QUADRATIC SOLUTIONS 497

where a is a 2n-dimensional column vector and b and c are n-dimensional columin vectors.

Finally, if di, , dn are n-dimensional column vectors, the symbol [di, d2 X * dn] will be used to denote the n by n matrix whose columns are di, ,dn .

THEOREM 1. Every solution of (1) has the form

X = [bi X * v bn2][Ci C * Xn]

where the column vectors bi and ci are the upper and lower halves of an eigenvector ai of M. Conversely, if a, ..., a are eigenvectors of M and [cl, ... * cn] is nonsingular, then

X = [bi, X bn][cl C nX , _

satisfies (1). THEOREM 2. If A and C are hermitian, a,, ... , an are eigenvectors of M

corresponding to eigenvalues X1, n, A, and Xi 5z? -Xj* for 1 _ i, j _ n, and if [ci , , * cn] is nonsingular, then

[bl, X bn] [ClX X Cn]_

is hermitian. THEOREM 3. Assume that A and C are positive semidefinite hermitian and

let a1, ... , an be eigenvectors of M corresponding to eigenvalue6 X1, Xi An In this case, (a) if A or C is nonsingular and

X = [b1, - bn][cl Cn1, is positive definite, then X1 , , Xn have positive real parts, and (b) if X1, * , Xn have positive real parts and [cl, ... * Cn] is nonsingular, then [bi X * bn] [cl X * * *, Cn]J1 is po6itive semidefinite.

Solutions of (1) may also be written in the form X = DE-, where the columns of D and E are the upper and lower halves of vectors which are linear combinations of n eigenvectors of M. This is hardly different from the representation given in Theorem 1 since if

D = [(,yn,bi + v + 'Yn1bn), *, (QY1nb + * + INybA)] and

E = [(eyiiC1 + + yNiCn), ..., (YinC1 + + -YnnCn)]

then

D = [bi, ,bn]G E = [cl, Cn]G,


498 JAMES E. POTTER

with G = [,yij] and, if E is nonsingular,

DE-' = [b1, v X bn][cl CnX X

On the other hand, Theorem 1 implies that, if D and E are made up of the upper and lower halves of nontrivial linear combinations of more than n eigenvectors of M corresponding to distinct eigenvalues, then DE-' does not satisfy (1). For suppose that a1', ... , am' with m > n are eigenvectors of M corresponding to distinct eigenvalues and that

X = DE-'

is a solution of (1) with

D = [bl', , bm']F

and

E = [cl', , cm.]F,

with F an m by n matrix having no row whose elements are all zeros. By Theorem 1,

X = HK 1;

where H and K are constructed from eigenvectors a1, ... , an. Then

[ai, . .. , a.] = [a,', ... , am']FE-'K.

Since eigenvectors corresponding to distinct eigenvalues are linearly independent, at least one row of FE'1K must contain all zero elements. The corresponding row of F would then be a null vector of E'1K under pre- multiplication which is impossible since E'1K is nonsingular.

In the applications mentioned above, the coefficient matrices A and C are positive semidefinite hermitian and a positive definite hermitian solu-

tion X is sought. If A and C are hermitian and [x] is an eigenvector of

M corresponding to the eigenvalue X, then [x] is an eigenvector of M* corresponding to the eigenvalue- X, and thuse -* is an eigenvalue of M. Hence M has at most n eigenvalues with positive real parts and, in view of Theorem 3, the eigenvectors to be used in forming a positive definite solution X are uniquely determined.

Proof of Theorem 1. To prove the first half of the theorem, suppose X is a solution of (1) and let

(2) G=CX-B*.


MATRIX QUADRATIC SOLUTIONS 499

By (1),

(3) XG = A + BX.

Let S transform G into its Jordan canonical form J, that is,

(4) S-'GS= J and let

(5) R = XS. Substituting (4) and (5) into (2) and (3) to eliminate G and X yields

RJ =BR + AS

and

SJ = CR - B*S.

Thus in

(6) Rs] =m R

J must be a diagonal matrix. For, if a,, ... a,n are the columns of [R] and J is not a diagonal matrix, then for some k,

0 = (M- XI)ak, and

ak = (M - XI)ak+l

Then ak is an eigenvector of M' corresponding to the eigenvalue X. Since M has a diagonal Jordan canonical form, its minimal polynomial is a product of distinct linear factors:

m (x) = (x- ) (X (- )(X-) Now

0 = m(M)ak[l = (X -1) ... (X - X,)ak and since X 5z? Xi for i = 1, ... , p, we conclude that ak = 0. But this is impossible since S is nonsingular.

It follows from the fact that J is diagoinal that a1, ..., an are eigenvectors of M and the desired result follows since X = RS-'. The second half of the theorem may be proved by carrying out the steps above in reverse order.


500 JAMES E. POTTER

Proof of Thcorem 2. Let

(7) P - [clX, c,n]*[bi, X * bn].

Since

(8) X = ([cl), , cn]l)*P[cl , * Cn, c] it is sufficient to prove that P is hermitian. Let T be the 2n by 2n matrix

T O :n -inl T In -n-|, where O? and In denote the n by n zero and identity matrices respectively, and let P = [Pjk]. Then

Pjk = cj*bk,

and

Pik - pkj = aj Tak. Since Xj* - Xk, we may write

Pjk - Pkj = (X*j + k) { Xj*a jTak + Xkaj*Tak}

= (Xj* + Xk)ylaj*(M*T + TM)ak

But

M*T + TM = 0,

and hence P is hermitian. Proof of Theorem 3. (a) Using the same notation as in the proof of

Theorem 1 we have

(9) G*X + XG = A + XCX.

Since the right-hand side of (9) is positive definite, it follows [7, p. 222] that the eigenvalues of G have positive real parts. The desired result follows since, by (6), the eigenvalues of G are X1, * *, Xn.

(b) We will make use of the matrix P defined in (7) above. In view of (8) it is sufficient to prove that P is positive semidefinite. Let U(t) denote the 2n by n matrix

U(t) = [exp (-X1t)a1, , exp (-Xnt)an].

Then

d U(t) = -MU(t). dt


MATRIX QUTADRATIC SOLUTIONS 501

Let L denote the 2n by 2n matrix

L 0 ?On -In [n:J

By (7) we have

P = U*(O)LU(O), and since U(t) 0 as t cc,

= fd { (t)LU(t) } dt (10)

= f U*(t){M*L + LM}U(t) dt. Since

M*L+LM=[ = Jo the integrand in (10) is positive semidefinite for every t and hence P is positive semidefinite.

REFERENCES [1] W. T. REID, A matrix differential equation of the Riccati type, Amer. J. Math., 68

(1946), pp. 237-246. [2] J. J. LEVIN, On the matrix Riccati equation, Proc. Amer. Math. Soc., 10(1959),

pp. 519-524. [3] R. L. STERNBERG AND H. KAUFMAN, Applications of the theory of systems of differ-

ential equations to multiple nonuniform transmission lines, J. Math. and Phys., 31(1952), pp. 244-252.

[4] R. E. KALMAN AND R. S. Bucy, New results in linear filtering and prediction theory, Trans. ASME Ser. D. J. Basic Engrg., 83D(1961), pp. 95-108.

[5] R. E. KALMAN, Contributions to the theory of optimal control, Proceedings of the Conference on Ordinary Differential Equations, Soc. Mat. Mexicana, Mexico City, 1959.

[6] N. WIENER, Extrapolation, Interpolation and Smoothing of Stationary Time Series, John Wiley, New York, 1949.

[7] F. R. GANTMACHER, Applications of the theory of matrices, Interscience, New York, 1959.


Article Contentsp. 496p. 497p. 498p. 499p. 500p. 501

Issue Table of ContentsSIAM Journal on Applied Mathematics, Vol. 14, No. 3 (May, 1966), pp. 417-639Front Matter [pp. ]A Note on Finite Convolution Operators [pp. 417-419]Self-Duality in Mathematical Programming [pp. 420-423]Note on the Krylov-Bogoliubov Method Applied to Linear Differential Equations [pp. 424-428]A Ricocheting Gradient Method for Nonlinear Optimization [pp. 429-445]On Volterra's Population Equation [pp. 446-452]On Partial Isometries in Finite-Dimensional Euclidean Spaces [pp. 453-467]A Block-Diagonalization Theorem for Systems of Linear Ordinary Differential Equations and Its Applications [pp. 468-475]A Generalization of the Radial Polynomials of F. Zernike [pp. 476-489]A Finite Series Solution of the Matrix Equation AX - XB = C [pp. 490-495]Matrix Quadratic Solutions [pp. 496-501]On the Determination of the Envelope of a Family of Epitrochoids with Applications [pp. 502-510]The Distribution of Products of Independent Random Variables [pp. 511-526]The Synthesis of Linear Dynamical Systems from Prescribed Weighting Patterns [pp. 527-549]On Directed Trees and Directed k-Trees of a Digraph and their Generation [pp. 550-560]A Distributional Hankel Transformation [pp. 561-576]Diffraction of a Scalar Wave by a Plane Screen [pp. 577-599]Self-Avoiding Paths and the Adjacency Matrix of a Graph [pp. 600-609]The Structure of Powers of Nonnegative Matrices: I. The Index of Convergence [pp. 610-639]Back Matter [pp. ]

Documents

1966 - Potter - Matrix Quadratic Solutions