半正定极因子在酉不变范数下的绝对与相对扰动界

The Absolute and Relative Perturbation Bounds for the Hermitian Positive Semidefinite Polar Factor under unitarily invariant norm

摘要: 设\bf A=\bf Q\bf H是矩阵\bf A\in \mathbb\bf C^m\times n的极分解, 其中\bf Q^*\bf Q=\bf I, \bf I为n阶单位矩阵, \bf 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.

Abstract: 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.