HERMITIAN POSITIVE DEFINITE SOLUTIONS OF THE MATRIX EQUATION X8+A*X-tA = Q
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In this paper, the Hermitian positive definite solutions of the matrix equation \({X^s} + {A^*}{X^{ - t}}A = Q\), where \(Q\) is an \(n\times n\) Hermitian positive definite matrix, \(A\) is an \(n\times n\) nonsingular complex matrix and \(s, t \in [1,\infty)\) are discussed. We find a matrix interval which contains all the Hermitian positive definite solutions of this equation. Also, a necessary and sufficient condition for the existence of these solutions is presented. Iterative methods for obtaining the maximal and minimal Hermitian positive definite solutions are proposed. The theoretical results are illustrated by numerical examples.
Abstract
In this paper, the Hermitian positive definite solutions of the matrix equation \({X^s} + {A^*}{X^{ - t}}A = Q\), where \(Q\) is an \(n\times n\) Hermitian positive definite matrix, \(A\) is an \(n\times n\) nonsingular complex matrix and \(s, t \in [1,\infty)\) are discussed. We find a matrix interval which contains all the Hermitian positive definite solutions of this equation. Also, a necessary and sufficient condition for the existence of these solutions is presented. Iterative methods for obtaining the maximal and minimal Hermitian positive definite solutions are proposed. The theoretical results are illustrated by numerical examples.
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