On the Hermitian Positive Definite Solutions of the Nonlinear Matrix Equation $X^s-A^*X^{-t}A=Q$ with Perturbation Estimates

Abstract:

      In this paper, the Hermitian positive definite solutions of the nonlinear matrix equation $X^s-A^*X^{-t}A=Q$ are studied, where $Q$ is a Hermitian positive definite matrix, $s$ and $t$ are positive integers. The existence of a Hermitian positive definite solution is proved. A sufficient condition for the equation to have a unique Hermitian positive definite solution is given. Some estimates of the Hermitian positive definite solutions are obtained. Moreover, two perturbation bounds for the Hermitian positive definite solutions are derived and the results are illustrated by some numerical examples.