On the Positive Definite Solutions of a Nonlinear Matrix Equation

Abstract

The positive definite solutions of the nonlinear matrix equation $X^s+A^{\ast }f(X)A=Q$ are discussed. A necessary and sufficient condition for the existence of positive definite solutions for this equation is derived. Then, the uniqueness of the Hermitian positive definite solution is studied based on an iterative method proposed in this paper. Lastly the perturbation analysis for this equation is discussed.