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

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