The iterative method for solving nonlinear matrix equation Xs + A∗X−tA = Q

Nonlinear matrix equation Xs + A∗X−tA = Q, where Q is an Hermitian positive definite matrix, A∗ is the conjugate transpose of the matrix A and both of s and t are positive integers, has widely applied background. In this paper, necessary conditions and sufficient conditions for the existence of Hermitian positive definite solutions of this equation are derived. An iterative method for obtaining Hermitian positive definite solutions is proposed. Moreover, a perturbation bound for the Hermitian positive definite solutions is given.