New Perturbation Bounds for the Unitary Polar Factor

Let A be an m n (m n) complex matrix. It is known that there is a unique polar decomposition A = QH, where Q Q = I, the n n identity matrix, and H is positive de nite, 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 Mathias's (SIAM J. Matrix Anal. Appl., 14(1993), 588{597.). For the non-square case, we also establish a bound for unitarily invariant norm, which has not been done in literature. Let A be an m n (m n) complex matrix. It is known that there are Q with orthonormal column vectors, i.e., Q Q = I , and a unique positive This material is based in part upon work supported by Argonne National Laboratory under grant No. 20552402 and the University of Tennessee through the Advanced Research Projects Agency under contract No. DAAL03-91-C-0047, by the National Science Foundation under grant No. ASC-9005933, and by the National Science Infrastructure grants No. CDA-8722788 and CDA-9401156.