On the Hermitian positive definite solutions of nonlinear matrix equation Xs + A∗X−tA = Q

Nonlinear matrix equation Xs + A∗X−tA = Q, where A, Q are n × n complex matrices with Q Hermitian positive definite, has widely applied background. In this paper, we consider the Hermitian positive definite solutions of this matrix equation with two cases: s ⩾ 1, 0 < t ⩽ 1 and 0 < s ⩽ 1, t ⩾ 1. We derive necessary conditions and sufficient conditions for the existence of Hermitian positive definite solutions for the matrix equation and obtain some properties of the solutions. We also propose iterative methods for obtaining the extremal Hermitian positive definite solution of the matrix equation. Finally, we give some numerical examples to show the efficiency of the proposed iterative methods.