关于非线性矩阵方程$X^s-A^*X^{-t}A=Q$的Hermitian正定解及其扰动估计

On the Hermitian Positive Definite Solutions of the Nonlinear Matrix Equation $X^s-A^*X^{-t}A=Q$ with Perturbation Estimates

中文关键词:  矩阵方程  Hermitian正定解  性质  存在性  扰动界.

英文关键词:matrix equation  Hermitian positive definite solution  property  existence  perturbation bound.

中文摘要:

      本文研究非线性矩阵方程$X^s-A^*X^{-t}A=Q$的Hermitian正定解, 其中$Q$是Hermitian正定矩阵, $s$和$t$为正整数. 文中证明了Hermitian正定解的存在性. 给出了方程存在唯一的Hermitian\\解的充分条件. 获得了解的若干估计. 此外, 还给出了Hermitian正定解的两个扰动界并用数值例子加以演示.

英文摘要:

      In this paper, the Hermitian positive definite solutions of the nonlinear matrix equation $X^s-A^*X^{-t}A=Q$ are studied, where $Q$ is a Hermitian positive definite matrix, $s$ and $t$ are positive integers. The existence of a Hermitian positive definite solution is proved. A sufficient condition for the equation to have a unique Hermitian positive definite solution is given. Some estimates of the Hermitian positive definite solutions are obtained. Moreover, two perturbation bounds for the Hermitian positive definite solutions are derived and the results are illustrated by some numerical examples.