Necessary and sufficient conditions for the existence of a positive definite solution of the matrix equation \(X+A^*X^{-1}A=Q\). (English) Zbl 0778.15008

Engwerda, Jacob C.; Ran, André C. M.; Rijkeboer, Arie L.

Linear Algebra Appl. 186, 255-275 (1993).

Necessary and sufficient conditions for the existence of a positive definite solution of the matrix equation \(X+A^*X^{-1}A=Q\), \(Q>0\) are proved using an analytic factorization approach. Both the real and the complex case are included. It is shown that the general case can be reduced to the case when \(Q=I\) and \(A\) is an invertible matrix. Algebraic recursive algorithms to compute the largest and the smallest solution of the equation are presented.
The number of solutions is described in terms of invariant subspaces for an invertible matrix \(A\). A relation to the theory of algebraic Riccati equations is outlined.

Reviewer: L.Bakule (Praha)

MSC:

15A24	Matrix equations and identities
15A23	Factorization of matrices
93C55	Discrete-time control/observation systems

Keywords:

symmetric factorizations; rational matrix-valued function; algebraic recursive algorithms; positive definite solution; matrix equation; analytic factorization; number of solutions; invariant subspaces; algebraic Riccati equations

References:

[1]	Alpay, D.; Gohberg, I., Unitary rational matrix functions, (Topics in Interpolation Theory of Rational Matrix-Valued Functions (1988), Birkhäuser: Birkhäuser Basel), 175-222, OT33 · Zbl 0651.47008
[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]	Ando, T., Topics on Operator Inequalities, Lecture Notes (1978), Hokkaido Univ: Hokkaido Univ Sapporo, Japan · Zbl 0388.47024
[4]	Ando, T., Structure of operators with numerical radius one, Acta Sci. Math. (Szeged), 34, 11-15 (1973) · Zbl 0258.47001
[5]	J.C. Engwerda, On the existence of a positive definite solution of the matrix equation \(XA^T X^{-1}AI\)Linear Algebra Appl.; J.C. Engwerda, On the existence of a positive definite solution of the matrix equation \(XA^T X^{-1}AI\)Linear Algebra Appl.
[6]	Gohberg, I.; Kaashoek, M. A.; Lay, D. C., Equivalence, linearization and decomposition of holomorphic operator functions, J. Funct. Anal., 28, 102-144 (1978) · Zbl 0384.47018
[7]	Gohberg, I.; Kaashoek, M. A.; Ran, A. C.M., Factorizations of and extensions to \(J\)-unitary rational matrix functions on the unit circle, Integral Equations Operator Theory, 15, 262-300 (1992) · Zbl 0792.47012
[8]	Gohberg, I.; Lancaster, P.; Rodman, L., Matrix Polynomials (1982), Academic · Zbl 0482.15001
[9]	Gohberg, I.; Lancaster, P.; Rodman, L., Matrices and Indefinite Scalar Products (1983), Birkhäuser · Zbl 0513.15006
[10]	Gohberg, I.; Lancaster, P.; Rodman, L., Invariant Subspaces of Matrices with Applications (1986), Wiley · Zbl 0608.15004
[11]	Hewer, G. A., An iterative technique for the computation of the steady state gains for the discrete optimal regulator, IEEE Trans. Automat. Control, AC-16, 382-383 (1971)
[12]	Ran, A. C.M.; Rodman, L., Stability of invariant maximal semidefinite subspaces. I, Linear Algebra Appl., 62, 51-86 (1984) · Zbl 0561.15001
[13]	Ran, A. C.M.; Rodman, L., Stability of invariant Lagrangian subspace I, Oper. Theory Adv. Appl., 32, 181-228 (1988) · Zbl 0658.47010
[14]	Ran, A. C.M.; Rodman, L., Stable hermitian solutions of discrete algebraic Riccati equations, Math. Control Signals Systems, 5, 165-193 (1992) · Zbl 0771.93059
[15]	Rosenblum, M.; Rovnyak, J., Hardy Classes and Operator Theory (1985), Oxford U.P · Zbl 0586.47020