矩阵方程X-A~*X^(-1)A=Q的Hermite正定解及其扰动分析冰

THE HERMITIAN POSITIVE DEFINITE SOLUTION AND ITS PERTURBATION ANALYSIS FOR THE MATRIX EQUATION X-A~*X^(-1)A=Q

考虑非线性矩阵方程X-A~*X^(-1)A=Q,其中A是n阶复矩阵,Q是n阶Hermite正定解,A~*是矩阵A的共轭转置.本文证明了此方程存在唯一的正定解,并推导出此正定解的扰动边界和条件数的显式表达式.以上结果用数值例子加以说明.

Consider the nonlinear matrix equation X-A＊x^-1A=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.