### Inequalities for the eigenvalues of the positive definite solutions of the nonlinear matrix equation

#### Abstract

In this manuscript, the Hermitian positive definite solutions of the matrix equation $X^s+A^*X^{-t}A=Q$ are considered,where $Q$ is a Hermitian positive definite matrix, $s$ and $t$ are positive integers. Bounds for the sum of eigenvalues of the solutions to the equation are given. Equivalent conditions for solutions of the equation are obtained. Eigenvalues of the solutions of the equation with the case $AQ = QA$ are investigated.

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