Perturbation of multiple eigenvalues of Hermitian matrices

Li, Ren-Cang and Nakatsukasa, Yuji and Truhar, Ninoslav and Wang, Wei-guo (2012) Perturbation of multiple eigenvalues of Hermitian matrices. [MIMS Preprint]

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Abstract

This paper is concerned with the perturbation of a multiple eigenvalue $\mu$ of the Hermitian matrix $A=\mbox{diag}(\mu I,A_{22})$ when it undergoes an off-diagonal perturbation $E$ whose columns have widely varying magnitudes. When some of $E$'s columns are much smaller than the others, some copies of $\mu$ are much less sensitive than any existing bound suggests. We explain this phenomenon by establishing individual perturbation bounds for different copies of $\mu$. They show that when $A_{22}-\mu I$ is definite the $i$th bound scales quadratically with the norm of the $i$th column, and in the indefinite case the bound is necessarily proportional to the product of $E$'s $i$th column norm and $E$'s norm. An extension to the generalized Hermitian eigenvalue problem is also presented.

Item Type: MIMS Preprint
Uncontrolled Keywords: Graded perturbation, multiple eigenvalue, generalized eigenvalue problem
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theory
MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis
Depositing User: Yuji Nakatsukasa
Date Deposited: 25 Jan 2012
Last Modified: 08 Nov 2017 18:18
URI: http://eprints.maths.manchester.ac.uk/id/eprint/1765

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