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.