On the change of the Weyr characteristics of matrix pencils after rank-one perturbations

TL;DR

This paper investigates how the Weyr characteristics of matrix pencils change under rank-one perturbations, providing new characterizations and bounds that extend previous results to more general settings.

Contribution

It introduces a novel characterization of Weyr characteristic changes using conjugate partitions and extends bounds to arbitrary rank-one perturbations over any algebraically closed field.

Findings

New characterization of Weyr characteristic change

Bounds for Weyr characteristic variation under rank-one perturbations

Results applicable over any algebraically closed field

Abstract

The change of the Kronecker structure of a matrix pencil perturbed by another pencil of rank one has been characterized in terms of the homogeneous invariant factors and the chains of column and row minimal indices of the initial and the perturbed pencils. We obtain here a new characterization in terms of the homogeneous invariant factors and the conjugate partitions of the corresponding chains of column and row minimal indices of both pencils. We also define the generalized Weyr characteristic of an arbitrary matrix pencil and obtain bounds for the change of it when the pencil is perturbed by another pencil of rank one. The results improve known results on the problem, hold for arbitrary perturbation pencils of rank one, and for any algebraically closed field.