The Hermitian positive definite solution of matrix equations X + A* X− A = I

In this paper, we discuss the Hermitian positive definite solutions of the nonlinear matrix equation X + A* X
^-1
A = I. we give some necessary and sufficient conditions for the existence of a Hermitian positive definite solution of equation(1.1). Based on them, we also present some properties of the coefficient matrix A are presented and two equivalent equation of equation(1.1) when the matrix equation has a Hermitian positive definite solution. And construct an iterative methods for obtaining the Hermitian positive definite solutions of the equation are constructed.