Generic Change of the Partial Multiplicities of Regular Matrix Pencils under Low-Rank Perturbations

Abstract

We describe the generic change of the partial multiplicities at a given eigenvalue $\lambda_0$ of a regular matrix pencil $A_0+\lambda A_1$ under perturbations with low normal rank. More precisely, if the pencil $A_0+\lambda A_1$ has exactly $g$ nonzero partial multiplicities at $\lambda_0$, then for most perturbations $B_0+\lambda B_1$ with normal rank $r<g$ the perturbed pencil $A_0+B_0+\lambda(A_1+B_1)$ has exactly $g-r$ nonzero partial multiplicities at $\lambda_0$, which coincide with those obtained after removing the largest $r$ partial multiplicities of the original pencil $A_0+\lambda A_1$ at $\lambda_0$. Though partial results on this problem had been previously obtained in the literature, its complete solution remained open.