## 2012.17: Perturbation of multiple eigenvalues of Hermitian matrices

2012.17:
Ren-Cang Li, Yuji Nakatsukasa, Ninoslav Truhar and Wei-guo Wang
(2012)
*Perturbation of multiple eigenvalues of Hermitian matrices.*

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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 |
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Uncontrolled Keywords: | Graded perturbation, multiple eigenvalue, generalized eigenvalue problem |

Subjects: | MSC 2000 > 15 Linear and multilinear algebra; matrix theory MSC 2000 > 65 Numerical analysis |

MIMS number: | 2012.17 |

Deposited By: | Yuji Nakatsukasa |

Deposited On: | 25 January 2012 |

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