# Generic change of the partial multiplicities of regular matrix pencils under low-rank perturbations Articles

## Overview

### published in

### publication date

- January 2016

### start page

- 823

### end page

- 835

### issue

- 3

### volume

- 37

### Digital Object Identifier (DOI)

### full text

### International Standard Serial Number (ISSN)

- 0895-4798

### Electronic International Standard Serial Number (EISSN)

- 1095-7162

### 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) + A(1) at lambda(0). Though partial results on this problem had been previously obtained in the literature, its complete solution remained open.

## Classification

### subjects

- Mathematics

### keywords

- regular matrix pencils; weierstrass canonical form; low-rank perturbations; matrix spectral perturbation theory; partial multiplicities; geometric approach