# A constructive proof of Pokrzywa's theorem about perturbations of matrix pencils

## TL;DR

This paper provides a new, constructive proof of Pokrzywa's theorem on perturbations of matrix pencils, simplifying the understanding of how small changes affect their canonical forms.

## Contribution

It offers a direct, constructive proof of Pokrzywa's theorem, reducing the problem to similarity and indecomposable cases, and explicitly calculating Kronecker forms in neighborhoods.

## Key findings

- Constructive proof of Pokrzywa's theorem.
- Explicit calculation of Kronecker forms near a given pencil.
- Reduction of the problem to similarity and indecomposable cases.

## Abstract

Our purpose is to give new proofs of several known results about perturbations of matrix pencils. Andrzej Pokrzywa (1986) described the closure of orbit of a Kronecker canonical pencil $A-\lambda B$ in terms of inequalities with pencil invariants. In more detail, Pokrzywa described all Kronecker canonical pencils $K-\lambda L$ such that each neighborhood of $A-\lambda B$ contains a pencil whose Kronecker canonical form is $K-\lambda L$. Another solution of this problem was given by Klaus Bongartz (1996) by methods of representation theory. We give a direct and constructive proof of Pokrzywa's theorem. We reduce its proof to the cases of matrices under similarity and of matrix pencils $P-\lambda Q$ that are direct sums of two indecomposable Kronecker canonical pencils. We calculate the Kronecker forms of all pencils in a neighborhood of such a pencil $P-\lambda Q$. In fact, we calculate the Kronecker forms of only those pencils that belong to a miniversal deformation of $P-\lambda Q$, which is sufficient since all pencils in a neighborhood of $P-\lambda Q$ are reduced to them by smooth strict equivalence transformations.

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Source: https://tomesphere.com/paper/1907.03213