We present eigenvalue bounds for perturbations of Hermitian matrices and express the change in eigenvalues in terms of a projection of the perturbation onto a particular eigenspace, rather than in terms of the full perturbation. The perturbations we consider are Hermitian of rank one, and Hermitian or non-Hermitian with norm smaller than the spectral gap of a specific eigenvalue. Applications include principal component analysis under a spiked covariance model, and pseudo-arclength continuation methods for the solution of nonlinear systems.
%0 Journal Article
%1 ipsen09
%A Ipsen, I. C. F.
%A Nadler, B.
%D 2009
%I Society for Industrial and Applied Mathematics
%J SIAM Journal on Matrix Analysis and Applications
%K eigenvalues perturbation
%N 1
%P 40-53
%R 10.1137/070682745
%T Refined Perturbation Bounds for Eigenvalues of Hermitian and Non-Hermitian Matrices
%V 31
%X We present eigenvalue bounds for perturbations of Hermitian matrices and express the change in eigenvalues in terms of a projection of the perturbation onto a particular eigenspace, rather than in terms of the full perturbation. The perturbations we consider are Hermitian of rank one, and Hermitian or non-Hermitian with norm smaller than the spectral gap of a specific eigenvalue. Applications include principal component analysis under a spiked covariance model, and pseudo-arclength continuation methods for the solution of nonlinear systems.
@article{ipsen09,
abstract = {We present eigenvalue bounds for perturbations of Hermitian matrices and express the change in eigenvalues in terms of a projection of the perturbation onto a particular eigenspace, rather than in terms of the full perturbation. The perturbations we consider are Hermitian of rank one, and Hermitian or non-Hermitian with norm smaller than the spectral gap of a specific eigenvalue. Applications include principal component analysis under a spiked covariance model, and pseudo-arclength continuation methods for the solution of nonlinear systems.},
added-at = {2014-01-26T20:10:04.000+0100},
author = {Ipsen, I. C. F. and Nadler, B.},
biburl = {https://www.bibsonomy.org/bibtex/2f9fd53aec06b47deca45d4e5a26df8e4/ytyoun},
doi = {10.1137/070682745},
interhash = {00a6aeaca3d80a32781b730a449f78eb},
intrahash = {f9fd53aec06b47deca45d4e5a26df8e4},
journal = {SIAM Journal on Matrix Analysis and Applications},
keywords = {eigenvalues perturbation},
month = jan,
number = 1,
pages = {40-53},
publisher = {Society for Industrial and Applied Mathematics},
timestamp = {2015-07-19T10:14:45.000+0200},
title = {Refined Perturbation Bounds for Eigenvalues of Hermitian and Non-Hermitian Matrices},
volume = 31,
year = 2009
}