On the existence of Hermitian positive definite solutions of the matrix equation \(X^s+A^*X^{-t}A=Q\). (English) Zbl 1143.15011

Duan, Xuefeng; Liao, Anping

Linear Algebra Appl. 429, No. 4, 673-687 (2008).

Summary: The existence of Hermitian positive definite solutions of the general nonlinear matrix equation \(X^s+A^*X^{-t}A=Q\) is studied systematically and deeply. A new estimate of Hermitian positive definite solutions is derived. Based on a fixed point theorem, some new sufficient conditions and new necessary conditions for the existence of Hermitian positive definite solutions are obtained. In the end, a necessary and sufficient condition for the existence of a Hermitian positive definite solution is proved.

MSC:

15A24	Matrix equations and identities
15A42	Inequalities involving eigenvalues and eigenvectors

Keywords:

nonlinear matrix equation; Hermitian positive definite solution; inequalities involving eigenvalues; fixed point theorem

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