Notes on the Hermitian Positive Definite Solutions of a Matrix Equation

الملخص EN

The nonlinear matrix equation, X-∑i=1mAi*XδiAi=Q, with -1≤δi<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.