On the Existence of Extremal Positive Definite Solutions of a Class of Nonlinear Matrix Equation

摘要

The nonlinear matrix equation X~r + Σ_(i=l)~m A_i~*X~(бi)A_i = Q is studied. A necessary condition for the existence of positive definite solutions of this equation is derived. Based on the Banach fixed point theorem, a sufficient condition for the existence of a unique positive definite solution of this equation is also derived. Iterative methods for obtaining the extremal (maximal–minimal) positive definite solutions of this equation are proposed.