Finite Rank Perturbations of Linear Relations and Matrix Pencils

TL;DR

This paper investigates how finite-rank perturbations affect the Jordan structure of linear relations and matrix pencils, revealing new bounds involving singular chains and extending known results to singular cases.

Contribution

It extends the analysis of Jordan structure deviations from operators to linear relations, introducing a sharp bound involving the factor n+1 and applying it to matrix pencils.

Findings

The difference in Jordan chains scales with n+1 for linear relations.

The bound is sharp, indicating the precise nature of the perturbation effect.

Application to matrix pencils includes new results for singular pencils.

Abstract

We elaborate on the deviation of the Jordan structures of two linear relations that are finite-dimensional perturbations of each other. We compare their number of Jordan chains of length at least $n$. In the operator case, it was recently proved that the difference of these numbers is independent of $n$ and is at most the defect between the operators. One of the main results of this paper shows that in the case of linear relations this number has to be multiplied by $n+1$ and that this bound is sharp. The reason for this behavior is the existence of singular chains. We apply our results to one-dimensional perturbations of singular and regular matrix pencils. This is done by representing matrix pencils via linear relations. This technique allows for both proving known results for regular pencils as well as new results for singular ones.