On the matrix equation \(X+A^ TX^{-1}A=I\). (English) Zbl 0863.15005

Zhan, Xingzhi; Xie, Jianjun

Linear Algebra Appl. 247, 337-345 (1996).

It is shown that the matrix equation \((*)\) \(X+A^TX^{-1}A=I\) has a positive definite solution \((X>0)\) if and only if \(A\) has a factorization \(A=W^TZ\), where the matrix \(W\) is nonsingular and the columns of \([W^T,Z^T]^T\) are orthonormal. In this case \(X=W^TW\). It is also proved that equation \((*)\) has a solution \(X>0\) if and only if there exist orthogonal matrices \(P\) and \(Q\) and diagonal matrices \(\Gamma>0\), \(\Sigma\geq 0\) with \(\Gamma^2+\Sigma^2=I\) such that \(A=P^T\Gamma Q\Sigma P\). Finally, it is shown that if \((*)\) has a solution \(X>0\) then the following relations are valid \(X-AA^T>0\), \(I-AA^T-A^TA>0\), \(r(A)\leq 1/2\), \(r(A+A^T)\leq 1\), \(r(A-A^T)\leq 1\), where \(r(A)\) is the spectral radius of \(A\).

Reviewer: M.M.Konstantinov (Sofia)

MSC:

15A24

Matrix equations and identities

Keywords:

matrix equation; positive definite solution; factorization

References:

[1]	Anderson, W. N.; Kleindorfer, G. B.; Kleindorfer, M. B.; 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 to \(X = A − BX^{−1}B∗\), Linear Algebra Appl., 134, 53-62 (1990) · Zbl 0702.15009
[3]	Anderson, W. N.; Morley, T. D.; Trapp, G. E., Ladder networks, fixed points, and the geometric mean, Circuits Systems Signal Process, 3, 259-268 (1983) · Zbl 0526.94017
[4]	Ando, T., Limit of cascade iteration of matrices, Numer. Funct. Anal. Optim., 21, 579-589 (1980)
[5]	Ando, T., Structure of operators with numerical radius one, Acta Sci. Math. (Szeged), 34, 11-15 (1973) · Zbl 0258.47001
[6]	Bucy, R. S., A priori bound for the Riccati equation, (Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Vol. III: Probability Theory (1972), Univ. of California Press: Univ. of California Press Berkeley), 645-656 · Zbl 0255.93025
[7]	Engwerda, J. C., On the existence of a positive definite solution of the matrix equation \(X = A^TX^{−1}A = I\), Linear Algebra Appl., 194, 91-108 (1993) · Zbl 0798.15013
[8]	Engwerda, J. C.; Ran, A. C.M.; 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
[9]	Golub, G. H.; Van Loan, C. F., Matrix Computations (1989), Johns Hopkins U.P: Johns Hopkins U.P Baltimore · Zbl 0733.65016
[10]	Green, W. L.; Kamen, E., Stabilization of linear systems over a commutative normed algebra with applications to spatially distributed parameter dependent systems, SIAM J. Control Optim., 23, 1-18 (1985) · Zbl 0564.93054
[11]	Horn, R. A.; Johnson, C. A., Matrix Analysis (1985), Cambridge U.P: Cambridge U.P Cambridge · Zbl 0576.15001
[12]	Ouellette, D. V., Schur complements and statistics, Linear Algebra Appl., 36, 187-295 (1981) · Zbl 0455.15012
[13]	Pusz, W.; Woronowitz, S. L., Functional calculus for sequilinear forms and the purification map, Rep. Math. Phys., 8, 159-170 (1975) · Zbl 0327.46032
[14]	Rosenblum, M.; Rovnyak, J., Hardy Classes and Operator Theory (1985), Oxford U.P · Zbl 0586.47020
[15]	Stewart, G. W.; Sun, J. G., Matrix Perturbation Theory (1990), Academic
[16]	Trapp, G. E., The Ricatti equation and the geometric mean, Contemp. Math., 47, 437-445 (1985)
[17]	Zabezyk, J., Remarks on the control of discrete time distributed parameter systems, SIAM J. Control, 12, 721-735 (1974) · Zbl 0254.93027
[18]	Zemanian, J., Non-uniform semi-infinite grounded grids, SIAM J. Appl. Math., 13, 770-788 (1982) · Zbl 0489.94029