半正定极因子在酉不变范数下的绝对与相对扰动界

设A=QH是矩阵ACm×n的极分解,其中Q*Q=I,I为n阶单位矩阵,H为n阶Hermite半正定矩阵.给出了任意扰动下Hermite半正定极因子在酉不变范数下的绝对与相对扰动界.对于满秩矩阵,绝对与相对扰动界具有最优性质.

THE ABSOLUTE AND RELATIVE PERTURBATION BOUNDS FOR THE HERMITIAN POSITIVE SEMIDEFINITE POLAR FACTOR UNDER UNITARILY INVARIANT NORM

Let ${\bf A}={\bf Q}{\bf H}$ be the polar decomposition of ${\bf A}\in {\bf C}^{m\times n}$, where ${\bf Q}^{*}{\bf Q}={\bf I}$ is the $n\times n$ identity, and ${\bf H}$ is Hermitian positive semi-definite. The absolute and relative perturbation bounds of Hermitian positive semi-definite polar factor for the matrices with different ranks are presented under any unitarily invariant norm. The absolute and relative bounds for matrices with full ranks are optimal.