THE HERMITIAN POSITIVE DEFINITE SOLUTION AND ITS PERTURBATION ANALYSIS FOR THE MATRIX EQUATION X-A~*X~(-1)A=Q
矩阵方程X-A~*X~(-1)A=Q的Hermite正定解及其扰动分析

Consider the nonlinear matrix equation X-A~*X~(-1)A=Q,where A,Q are n×n complex matrices with Q Hermitian positive definite and A~* denotes the conjugate transpose of a matrix A.This paper shows there exists a unique positive definite solution to the equation. The perturbation bounds for the Hermitian positive definite solution to the matrix equation are derived,explicit expressions of the condition number for the Hermitian positive definite solution are obtained and the backward error of an approximate solution to the Hermitian positive definite solution is evaluated.The results are illustrated by numerical examples.