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

Abstract

We consider the nonlinear matrix equation X=Q+A∗(X^−C)−1A, where Q is positive definite, C is positive semidefinite, and X^ is the block diagonal matrix defined by X^=diag(X,X,…,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.