The Hermitian positive definite solution of a type of nonlinear matrix equations X + A*X -1A = I

In this paper, the Hermitian positive definite solutions of the nonlinear matrix equation X + A*X
^-1
A = I are studied. Suppose X is a positive definite solution of this equation, we discuss the relation between X and A by the eigenvalue and eigenvector of X and A respectively, and the property of numerical range of A. An iterative method for obtaining the Hermitian positive definite solutions of the equation are constructed.