Notes on the hermitian positive definite solutions of a matrix equation

The nonlinear matrix equation,x - ∑ i = 1 m a ix δ i a i = q,with - 1 ≤ δ i < 0 is investigated. a fixed point theorem in partially ordered sets is proved. and then,by means of this fixed point theorem,the existence of a unique hermitian positive definite solution for the matrix equation is derived. some properties of the unique hermitian positive definite solution are obtained. a residual bound of an approximate solution to the equation is evaluated. the theoretical results are illustrated by numerical examples. © 2014 jing li and yuhai zhang.