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.