engleski

Perturbation of multipleeigenvalues of Hermitian matrices

This paper is concerned with the perturbation of the multiple eigenvalue $\mu$ of Hermitian matrices of the form $A=\mbox{; ; ; ; diag}; ; ; ; (\mu I, A_{; ; ; ; 22}; ; ; ; )$, when the matrix 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 insensitive than any existing bound suggests. We explain this phenomenon by showing 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 generalized Hermitian eigenvalue problems is presented.

Graded perturbation; multiple eigenvalue; generalized eigenvalue problem

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