The nonlinear matrix equation, with 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.
Jing Li. Yuhai Zhang. "Notes on the Hermitian Positive Definite Solutions of a Matrix Equation." J. Appl. Math. 2014 1 - 8, 2014. https://doi.org/10.1155/2014/128249