# Relative Perturbation Theory: (II) Eigenspace Variations

### Ren-Cang Li

###
EECS Department

University of California, Berkeley

Technical Report No. UCB/CSD-94-856

December 1994

### http://www.eecs.berkeley.edu/Pubs/TechRpts/1994/CSD-94-856.pdf

In this paper, we consider how eigenspaces of a Hermitian matrix
*A* change when it is perturbed to ~
*A* =
*D* *
*AD* and how singular values of a (nonsquare) matrix
*B* change when it is perturbed to ~
*B* =
*D*1
*BD*2, where
*D*,
*D*1 and
*D*2 are assumed to be close to identity matrices of suitable dimensions, or either
*D*1 or
*D*2 close to some unitary matrix. We have been able to generalize well-known Davis-Kahan sin theta theorems. As applications, we obtained bounds for perturbations of graded matrices.

BibTeX citation:

@techreport{Li:CSD-94-856, Author = {Li, Ren-Cang}, Title = {Relative Perturbation Theory: (II) Eigenspace Variations}, Institution = {EECS Department, University of California, Berkeley}, Year = {1994}, Month = {Dec}, URL = {http://www.eecs.berkeley.edu/Pubs/TechRpts/1994/5482.html}, Number = {UCB/CSD-94-856}, Abstract = {In this paper, we consider how eigenspaces of a Hermitian matrix <i>A</i> change when it is perturbed to ~<i>A</i> = <i>D</i> * <i>AD</i> and how singular values of a (nonsquare) matrix <i>B</i> change when it is perturbed to ~<i>B</i> = <i>D</i>1<i>BD</i>2, where <i>D</i>, <i>D</i>1 and <i>D</i>2 are assumed to be close to identity matrices of suitable dimensions, or either <i>D</i>1 or <i>D</i>2 close to some unitary matrix. We have been able to generalize well-known Davis-Kahan sin theta theorems. As applications, we obtained bounds for perturbations of graded matrices.} }

EndNote citation:

%0 Report %A Li, Ren-Cang %T Relative Perturbation Theory: (II) Eigenspace Variations %I EECS Department, University of California, Berkeley %D 1994 %@ UCB/CSD-94-856 %U http://www.eecs.berkeley.edu/Pubs/TechRpts/1994/5482.html %F Li:CSD-94-856