## On the matrix equation $$X+A^ TX^{-1}A=I$$.(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 $$\Gamma>0$$, $$\Sigma\geq 0$$ 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 $$A$$.

### MSC:

 15A24 Matrix equations and identities

### Keywords:

matrix equation; positive definite solution; factorization
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### 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
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