Extension of Matrix Pencil Reduction to Abelian Categories

TL;DR

This paper extends the classical Kronecker decomposition of matrix pencils to pairs of morphisms in abelian categories, providing new proof techniques and invariants that generalize well-known results.

Contribution

It generalizes the matrix pencil reduction and Kronecker decomposition to abelian categories, introducing new invariants and proof methods applicable beyond vector spaces.

Findings

Partial decomposition results hold in abelian categories

Reduction processes commute and produce invariant sequences

Relation between invariants and resolvent set in module categories

Abstract

Matrix pencils, or pairs of matrices, are used in a variety of applications. By the Kronecker decomposition Theorem, they admit a normal form. This normal form consists of four parts, one part based on the Jordan canonical form, one part made of nilpotent matrices, and two other dual parts, which we call the observation and control part. The goal of this paper is to show that large portions of that decomposition are still valid for pairs of morphisms of modules or abelian groups, and more generally in any abelian category. % This gives a new perspective even in the vector space case, as we have to use radically new proof techniques to work on abelian categories. In the vector space case, we recover the full Kronecker decomposition theorem. The main technique is that of reduction, which extends readily to the abelian category case. Reductions naturally arise in two flavours, which are dual to each other. There are a number of properties of those reductions which extend remarkably from the vector space case to abelian categories. First, both types of reduction commute. Second, at each step of the reduction, one can compute three sequences of invariant spaces (objects in the category), which generalize the Kronecker decomposition into nilpotent, observation and control blocks. These sequences indicate whether the system is reduced in one direction or the other. In the category of modules, there is also a relation between these sequences and the resolvent set of the pair of morphisms, which generalizes the regular pencil theorem. We also indicate how this allows to define invariant subspaces in the vector space case, and study the notion of strangeness as an example.