Hermitian solutions of the equation \(X=Q+NX^{-1}N^*\). (English) Zbl 0876.15011

Ferrante, Augusto; Levy, Bernard C.

Linear Algebra Appl. 247, 359-373 (1996).

The authors consider the set \({\mathcal X}\) of Hermitian matrices \(X\) which satisfy \(X = Q + NX^{-1}N^*\) where \(Q\) and \(N\) are specified \(n \times n\) matrices over \({\mathbb C}\) and \(Q\) is Hermitian positive definite. They show that that there is a one-to-one correspondence between the elements \(X\) in \({\mathcal X}\) and the factorizations of the matrix polynomial \(N(s,t) := Qst + Ns^{2} - N^*t^{2}\) in the form \(N(s,t) = (tI + sM^*)X(sI - tM)\) for some \(M\). They also show that under the usual ordering for Hermitian matrices \({\mathcal X}\) has a unique maximal element \(X_{+}\) and, if \(N\) is nonsingular, a unique minimal element \(X_{-}\). Moreover, \(X_{+}\) can be computed as the limit of the sequence \(\{X_{k}\}\) where \(X_{0} := Q\) and \(X_{k+1} := Q + NX^{-1}_{k}N^*\).

Reviewer: J.D.Dixon (Ottawa)

MSC:

15A24

Matrix equations and identities

Keywords:

matrix equation; Hermitian matrices; matrix polynomial

References:

[1]	Anderson, W. N.; Kleindorfer, G. B.; Kleindorfer, P. R.; Woodroofe, M. B., Consistent estimates of the parameters of a linear system, Ann. Math. Statist., 40, 2064-2075 (1969) · Zbl 0213.20703
[2]	Anderson, W. N.; Morley, T. D.; Trapp, G. E., Positive solutions of the matrix equation \(X = A − B∗X^{−1}B\), Linear Algebra Appl., 134, 53-62 (1990) · Zbl 0702.15009
[3]	Demmel, J. W.; Kågstrom, B., Computing stable eigendecompositions of matrix pencils, Linear Algebra Appl., 88/89, 139-186 (1987) · Zbl 0627.65032
[4]	Engwerda, J. C., On the existence of a positive definite solution of the matrix equation \(X + A∗X^{−1}A = I\), Linear Algebra Appl., 194, 91-108 (1993) · Zbl 0798.15013
[5]	Engwerda, J. C.; Ran, A. C.; Rijkeboer, A. L., Necessary and sufficient conditions for the existence of a positive definite solution of the matrix equation \(X + A∗X^{−1}A = Q\), Linear Algebra Appl., 186, 255-275 (1993) · Zbl 0778.15008
[6]	Golub, G. H.; Van Loan, C. F., Matrix Computations (1989), Johns Hopkins U.P: Johns Hopkins U.P Baltimore · Zbl 0733.65016
[7]	Horn, R. A.; Johnson, C. R., Topics in Matrix Analysis (1991), Cambridge U.P · Zbl 0729.15001
[8]	Levy, B. C., Regular and reciprocal multivariate stationary Gaussian reciprocal processes over Z are necessarily Markov, J. Math. Systems Estim. Control, 2, 133-154 (1992)
[9]	Levy, B. C.; Frezza, R.; Krener, A. J., Modeling and estimation of discrete-time Gaussian reciprocal processes, IEEE Trans. Automat. Control, 35, 1013-1023 (1990) · Zbl 0709.60043
[10]	Stewart, G., On the sensitivity of the eigenvalue problem Ax = λBx, SIAM J. Numer. Anal., 9, 669-696 (1972) · Zbl 0252.65026