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