Necessary and Sufficient Conditions for the Existence of a Hermitian Positive Definite Solution of a Type of Nonlinear Matrix Equations

We study the Hermitian positive definite solutions of the nonlinear matrix equation X+A∗X−2A=I, where A is an n×n nonsingular matrix. Some necessary and sufficient conditions for the existence of a Hermitian positive definite solution of this equation are given. However, based on the necessary and sufficient conditions, some properties and the equivalent equations of X+A∗X−2A=I are presented while the matrix equation has a Hermitian positive definite solution.