### New Perturbation Bounds for the Unitary Polar Factor

**Authors:**

Li, Ren-Cang
**Technical Report Identifier:** CSD-94-852
**December 1994**

CSD-94-852.pdf

CSD-94-852.ps

**Abstract:** Let *A* be an *m* x *n* (*m* >= *n*) complex matrix. It is known that there is a unique polar decomposition *A* = *QH*, where *Q***Q* = *I*, the *n* x *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 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.