On Hermitian Positive Definite Solution of Nonlinear Matrix Equation $X+A^* X^{-2}A=Q$

Abstract

Based on the fixed-point theory, we study the existence and the uniqueness of the maximal Hermitian positive definite solution of the nonlinear matrix equation $X+A^*X^{-2}A=Q$, where $Q$ is a square Hermitian positive definite matrix and $A^*$ is the conjugate transpose of the matrix $A$. We also demonstrate some essential properties and analyze the sensitivity of this solution. In addition, we derive computable error bounds about the approximations to the maximal Hermitian positive definite solution of the nonlinear matrix equation $X+A^*X^{-2}A=Q$. At last, we further generalize these results to the nonlinear matrix equation $X+A^*X^{-n}A=Q$, where $n \ge 2$ is a given positive integer.