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

We describe the generic change of the partial multiplicities at a given eigenvalue λ0 of a regular matrix pencil A0 + λA1 under perturbations with low normal rank. More precisely, if the pencil A0 + λA1 has exactly g nonzero partial multiplicities at λ0, then for most perturbations B0 + λB1 with normal rank r < g the perturbed pencil A0 + B0 + λ(A1 + B1) has exactly g − r nonzero partial multiplicities at λ0, which coincide with those obtained after removing the largest r partial multiplicities of the original pencil A0 + λA1 at λ0. Though partial results on this problem had been previously obtained in the literature, its complete solution remained open.