On characteristic invariants of matrix pencils and linear relations

TL;DR

This paper explores the connection between linear relations and matrix pencils through Weyr characteristics, linking them to Kronecker forms and analyzing invariants under rank-one perturbations.

Contribution

It introduces Weyr characteristics for linear relations and establishes their correspondence with Kronecker canonical forms of matrix pencils.

Findings

Weyr characteristic relates to Kronecker form

Invariant characteristics estimated under rank-one perturbations

Provides a new perspective on matrix pencil invariants

Abstract

The relationship between linear relations and matrix pencils is investigated. Given a linear relation, we introduce its Weyr characteristic. If the linear relation is the range (or the kernel) representation of a given matrix pencil, we show that there is a correspondence between this characteristic and the Kronecker canonical form of the pencil. This relationship is exploited to obtain estimations on the invariant characteristics of matrix pencils under rank one perturbations.