Electrical Engineering
      and Computer Sciences

Electrical Engineering and Computer Sciences

COLLEGE OF ENGINEERING

UC Berkeley

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 = D1 AD2 and how singular values of a (nonsquare) matrix B change when it is perturbed to ~ B = D1 BD2, where D1 and D2 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