Existence and Uniqueness of the Positive Definite Solution for the Matrix Equation X = Q + A ∗ ( X ^ − C ) − 1 A

Abstract

We consider the nonlinear matrix equation $X=Q+A^{\ast }{(\widehat{X}-C)}^{-1}A$, where $Q$ is positive definite, $C$ is positive semidefinite, and $\widehat{X}$ is the block diagonal matrix defined by $\widehat{X}=\text{d}\text{i}\text{a}\text{g}(X,X,\dots ,X)$. We prove that the equation has a unique positive definite solution via variable replacement and fixed point theorem. The basic fixed point iteration for the equation is given.