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On the Hermitian positive definite solutions of nonlinear matrix equation ${X}^{s}+{A}^{*}{X}^{-t}A=Q$. (English) Zbl 1201.15005
Authors’ abstract: The nonlinear matrix equation ${X}^{s}+{A}^{*}{X}^{-t}A=Q$, where $A,Q$ are $n×n$ complex matrices with $Q$ Hermitian positive definite, has widely applied background. In this paper, we consider the Hermitian positive definite solutions of this matrix equation with two cases: $s⩾1,0 and $0. We derive necessary conditions and sufficient conditions for the existence of Hermitian positive definite solutions for the matrix equation and obtain some properties of the solutions. We also propose iterative methods for obtaining the extremal Hermitian positive definite solution of the matrix equation. Finally, we give some numerical examples to show the efficiency of the proposed iterative methods.
##### MSC:
 15A24 Matrix equations and identities 65F30 Other matrix algorithms 15B48 Positive matrices and their generalizations; cones of matrices 15B57 Hermitian, skew-Hermitian, and related matrices