On Positive Definite Solutions of Nonlinear Matrix Equation X - A*X(-1)A = I

Abstract

This paper is concerned with the positive definite solution of the nonlinear matrix equation X - A*X(-1)A = I where A is a given complex matrix. The existence and uniqueness of the positive definite solution are established and several efficient numerical algorithms for obtaining the unique positive definite solution are also derived. Numerical examples are worked out to illustrate the effectiveness of the proposed algorithms.