Some new perturbation bounds for the generalized polar decomposition (Q1770935)

The generalized polar decomposition of an \(m\times n\) complex matrix \(A\) of rank \(r\) is \(A=QH\), where \(Q=U_1V_1^\star\), \(H=V_1\Sigma_1V_1^\star\), and \[ A=U\Sigma V^\star=(U_1,U_2)\begin{pmatrix} \Sigma_1&0\\ 0&0\end{pmatrix} (V_1,V_2)^\star=U_1\Sigma_1V_1^\star, \] \(\Sigma_1=\text{ diag}\,(\sigma_1,\ldots,\sigma_r)\) with \(\sigma_1\geq\sigma_2\geq\cdots\geq\sigma_r>0\). The authors introduce a new measure of perturbation and give formulas for the additive and multiplicative perturbation bounds for the subunitary polar factor \(Q\).