Necessary and sufficient conditions for the existence of a positive definite solution of the matrix equation $X+{A}^{*}{X}^{-1}A=Q$.

*(English)*Zbl 0778.15008Necessary 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 systems |