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On the existence of a positive definite solution of the matrix equation X+A T X -1 A=I. (English) Zbl 0798.15013

First, the author demonstrates that the problem described in the title is equivalent to the recursion problem n, is X n >AA T if X n+1 =I-A T X n -1 A with X 0 =I. From this follow not only necessary solvability conditions in terms of the spectral radius of A, A+A T and A-A T , respectively, but also an algorithm for the numerical computation of the solution. For normal matrices A one of these conditions is necessary and sufficient. Moreover, it is demonstrated that the matrix A need not be regular and how this equation can be used to solve an indefinite linear/quadratic, discrete-time optimal control problem.

The paper is well-organized, and easy to read.

Reviewer: I.Troch (Wien)

MSC:
15A24Matrix equations and identities
93C55Discrete-time control systems
65F30Other matrix algorithms
49N10Linear-quadratic optimal control problems