Necessary and sufficient conditions for the existence of a Hermitian positive definite solution of a type of nonlinear matrix equations (Q1036420)

Summary: We study the Hermitian positive definite solutions of the nonlinear matrix equation \(X+A^{\ast }X^{ - 2}A=I\), where \(A\) is an \(n\times n\) 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 \(X+A^{\ast }X^{ - 2}A=I\) are presented while the matrix equation has a Hermitian positive definite solution.