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: | https://eprints.maths.manchester.ac.uk/id/eprint/1765 |
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