# Fixed rank perturbations of regular matrix pencils

## TL;DR

This paper characterizes the structure of regular matrix pencils under fixed rank perturbations, extending previous bounded rank results and providing conditions for prescribed determinants.

## Contribution

It introduces a new characterization of regular matrix pencils under fixed rank perturbations, generalizing prior bounded rank perturbation results.

## Key findings

- Provides necessary and sufficient conditions for fixed rank perturbations with prescribed determinants.
- Extends the structure theory of matrix pencils to fixed rank perturbations.
- Results applicable over fields with sufficient elements.

## Abstract

A characterization of the structure of a regular matrix pencil obtained by a bounded rank perturbation of another regular matrix pencil has been recently obtained. The result generalizes the solution for the bounded rank perturbation problem of a square constant matrix. When comparing the fixed rank perturbation problem of a constant matrix with the bounded rank perturbation problem we realize that both problems are of different nature; the first one is more restrictive. In this paper we characterize the structure of a regular matrix pencil obtained by a fixed rank perturbation of another regular matrix pencil. We apply the result to find necessary and sufficient conditions for the existence of a fixed rank perturbation such that the perturbed pencil has a prescribed determinant. The results hold over fields with sufficient number of elements.