# Relative Perturbation Theory: (I) Eigenvalue Variations

### Ren-Cang Li

###
EECS Department

University of California, Berkeley

Technical Report No. UCB/CSD-94-855

December 1994

### http://www.eecs.berkeley.edu/Pubs/TechRpts/1994/CSD-94-855.pdf

In this paper, we consider how eigenvalues of a matrix
*A* change when it is perturbed to ~
*A* =
*D*1
*AD*2 and how singular values of a (nonsquare) matrix
*B* change when it is perturbed to ~
*B* =
*D*1
*BD*2, where
*D*1 and
*D*2 are assumed to be close to unitary matrices of suitable dimensions. We have been able to generalize many well-known perturbation theorems, including Hoffman-Wielandt theorem and Weyl-Lidskii theorem. As applications, we obtained bounds for perturbations of graded matrices in both singular value problems and nonnegative definite Hermitian eigenvalue problems.

BibTeX citation:

@techreport{Li:CSD-94-855, Author = {Li, Ren-Cang}, Title = {Relative Perturbation Theory: (I) Eigenvalue Variations}, Institution = {EECS Department, University of California, Berkeley}, Year = {1994}, Month = {Dec}, URL = {http://www.eecs.berkeley.edu/Pubs/TechRpts/1994/5481.html}, Number = {UCB/CSD-94-855}, Abstract = {In this paper, we consider how eigenvalues of a matrix <i>A</i> change when it is perturbed to ~<i>A</i> = <i>D</i>1<i>AD</i>2 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>1 and <i>D</i>2 are assumed to be close to unitary matrices of suitable dimensions. We have been able to generalize many well-known perturbation theorems, including Hoffman-Wielandt theorem and Weyl-Lidskii theorem. As applications, we obtained bounds for perturbations of graded matrices in both singular value problems and nonnegative definite Hermitian eigenvalue problems.} }

EndNote citation:

%0 Report %A Li, Ren-Cang %T Relative Perturbation Theory: (I) Eigenvalue Variations %I EECS Department, University of California, Berkeley %D 1994 %@ UCB/CSD-94-855 %U http://www.eecs.berkeley.edu/Pubs/TechRpts/1994/5481.html %F Li:CSD-94-855