Extension of Matrix Pencil Reduction to Abelian Categories
Olivier Verdier

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…
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