New Perturbation Bounds for the Unitary Polar Factor
Ren‐Cang Li · SIAM Journal on Matrix Analysis and Applications · 1995
Let A be an $m \times n\, (m \geq n)$ complex matrix. It is known that there is a unique polar decomposition$A = QH$, where $Q^ * Q = I$, the $n \times n$ identity matrix, and H is positive definite, provided A has full column rank. This note addresses the following question; How much may Q change if A is perturbed? For the square case $m = n$ our bound, which is valid for any unitarily invariant norm, is sharper and simpler than that of Mathias [SIAM J. Matrix Anal. Appl., 14 (1993), pp. 588–597]. For the nonsquare case, a bound is also established for unitarily invariant norm, which has not been done in the literature.