# Relative Perturbation Bounds for the Unitary Polar Factor

### http://www2.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 = D1 BD2? Here D1 and D2 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 D1 and D2 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://www2.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://www2.eecs.berkeley.edu/Pubs/TechRpts/1994/5480.html
%F Li:CSD-94-854
```