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On the matrix equation $X+A\sp TX\sp{-1}A=I$. (English) Zbl 0863.15005
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 $\Gamma>0$, $\Sigma\ge 0$ 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 $X-AA^T>0$, $I-AA^T-A^TA>0$, $r(A)\le 1/2$, $r(A+A^T)\le 1$, $r(A-A^T)\le 1$, where $r(A)$ is the spectral radius of $A$.

MSC:
15A24Matrix equations and identities
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[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-1B\ast. 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, 645-656 (1972) · Zbl 0255.93025
[7] Engwerda, J. C.: On the existence of a positive definite solution of the matrix equation X = ATX-1A = 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\astx-1A = Q. Linear algebra appl. 186, 255-275 (1993) · Zbl 0778.15008
[9] Golub, G. H.; Van Loan, C. F.: Matrix computations. (1989) · 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) · 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) · Zbl 0586.47020
[15] Stewart, G. W.; Sun, J. G.: Matrix perturbation theory. (1990) · Zbl 0706.65013
[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