### Relative Perturbation Bounds for the Unitary Polar Factor

**Authors:**

Li, Ren-Cang
**Technical Report Identifier:** CSD-94-854
**December 1994**

CSD-94-854.pdf

CSD-94-854.ps

**Abstract:** Let *B* be an *m* x *n* (*m* >= *n*) complex matrix. It is known that there is a unique polar decomposition *B* = *QH*, where *Q***Q* = *I*, the *n* x *n* identity matrix, and *H* is positive definite, provided *B* has full column rank. This paper addresses the following question: how much may *Q* change if *B* is perturbed to ~*B* = *D*1*BD*2? Here *D*1 and *D*2 are two nonsingular matrices and close to the identities of suitable dimensions.

Known perturbation bounds for complex matrices indicate that in the worst case, the change in *Q* is proportional to the reciprocal of the smallest singular value of *B*. In this paper, we will prove that for the above mentioned perturbations to *B*, the change in *Q* is bounded only by the distances from *D*1 and *D*2 to identities!

As an application, we will consider perturbations for one-side scaling, i.e., the case when *G* = *D* * *B* is perturbed to ~*G* = *D* * ~*B*, where *D* is usually a nonsingular diagonal scaling matrix but for our purpose we do not have to assume this, and *B* and ~*B* are nonsingular.