---
abstract: 'We solve the problem of characterizing the Kronecker structure of a matrix pencil obtained by a rank-one perturbation of another matrix pencil. The results hold over arbitrary fields.'
author:
- 'Itziar Baragaña[^1], Alicia Roca[^2]'
bibliography:
- 'referencesreg.bib'
title: 'Rank-one perturbations of matrix pencils'
---

matrix pencils, Kronecker structure, rank perturbation

15A21, 15A22, 47A55

Introduction
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Given a matrix pencil $A(s)=A_0+sA_1 \in {\mathbb F}^{n\times m}$, the rank perturbation problem consists in characterizing the Kronecker structure of $A(s)+P(s)$, where $P(s)$ is a matrix pencil of bounded rank.

The Kronecker structure of a matrix pencil is determined by the complete system of invariants for the strict equivalence of matrix pencils, i.e., the invariant factors, infinite elementary divisors, and row and column minimal indices. For regular matrix pencils the Kronecker structure is known as the Weierstrass structure and is determined only by the invariant factors and the infinite elementary divisors. In particular, the Jordan structure of a square matrix is defined by the Weierstrass structure of the associated characteristic pencil, which is a regular pencil without infinite elementary divisors. Analogously, the feedback invariants of a pair of matrices, i.e., the invariant factors and the column (or row) minimal indices are the Kronecker 