# Relative Perturbation Bounds for the Unitary Polar Factor

### Ren-Cang Li

###
EECS Department

University of California, Berkeley

Technical Report No. UCB/CSD-94-854

December 1994

### http://www.eecs.berkeley.edu/Pubs/TechRpts/1994/CSD-94-854.pdf

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.

BibTeX citation:

@techreport{Li:CSD-94-854, Author = {Li, Ren-Cang}, Title = {Relative Perturbation Bounds for the Unitary Polar Factor}, Institution = {EECS Department, University of California, Berkeley}, Year = {1994}, Month = {Dec}, URL = {http://www.eecs.berkeley.edu/Pubs/TechRpts/1994/5480.html}, Number = {UCB/CSD-94-854}, Abstract = {Let <i>B</i> be an <i>m</i> x <i>n</i> (<i>m</i> >= <i>n</i>) complex matrix. It is known that there is a unique polar decomposition <i>B</i> = <i>QH</i>, where <i>Q</i>*<i>Q</i> = <i>I</i>, the <i>n</i> x <i>n</i> identity matrix, and <i>H</i> is positive definite, provided <i>B</i> has full column rank. This paper addresses the following question: how much may <i>Q</i> change if <i>B</i> is perturbed to ~<i>B</i> = <i>D</i>1<i>BD</i>2? Here <i>D</i>1 and <i>D</i>2 are two nonsingular matrices and close to the identities of suitable dimensions. <p>Known perturbation bounds for complex matrices indicate that in the worst case, the change in <i>Q</i> is proportional to the reciprocal of the smallest singular value of <i>B</i>. In this paper, we will prove that for the above mentioned perturbations to <i>B</i>, the change in <i>Q</i> is bounded only by the distances from <i>D</i>1 and <i>D</i>2 to identities! <p>As an application, we will consider perturbations for one-side scaling, i.e., the case when <i>G</i> = <i>D</i> * <i>B</i> is perturbed to ~<i>G</i> = <i>D</i> * ~<i>B</i>, where <i>D</i> is usually a nonsingular diagonal scaling matrix but for our purpose we do not have to assume this, and <i>B</i> and ~<i>B</i> are nonsingular.} }

EndNote citation:

%0 Report %A Li, Ren-Cang %T Relative Perturbation Bounds for the Unitary Polar Factor %I EECS Department, University of California, Berkeley %D 1994 %@ UCB/CSD-94-854 %U http://www.eecs.berkeley.edu/Pubs/TechRpts/1994/5480.html %F Li:CSD-94-854