# 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]

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.