On bundles of matrix pencils under strict equivalence

TL;DR

This paper investigates the topological properties of bundles of matrix pencils under strict equivalence, providing explicit characterizations of their closure relations and proving their openness in the standard topology.

Contribution

It offers a formal analysis of the inclusion relations between bundle closures and establishes that certain bundles are open in their closures, addressing open questions in the topology of matrix pencil bundles.

Findings

Bundles of matrices under similarity are open in their closure.

Bundles of matrix polynomials are open in their closure.

Provides explicit characterization of inclusion relations between bundle closures.

Abstract

Bundles of matrix pencils (under strict equivalence) are sets of pencils having the same Kronecker canonical form, up to the eigenvalues (namely, they are an infinite union of orbits under strict equivalence). The notion of bundle for matrix pencils was introduced in the 1990's, following the same notion for matrices under similarity, introduced by Arnold in 1971, and it has been extensively used since then. Despite the amount of literature devoted to describing the topology of bundles of matrix pencils, some relevant questions remain still open in this context. For example, the following two: (a) provide a characterization for the inclusion relation between the closures (in the standard topology) of bundles; and (b) are the bundles open in their closure? The main goal of this paper is providing an explicit answer to these two questions. In order to get this answer, we also review and/or formalize some notions and results already existing in the literature. We also prove that bundles of matrices under similarity, as well as bundles of matrix polynomials (defined as the set of $m\times n$ matrix polynomials of the same grade having the same spectral information, up to the eigenvalues) are open in their closure.