On bundle closures of matrix pencils and matrix polynomials

TL;DR

This paper investigates the structure of bundles of matrix polynomials, proving that their closures are unions of smaller-dimensional bundles and providing formulas for their dimensions based on eigenstructure characteristics.

Contribution

It establishes that the closure of a bundle of matrix polynomials includes finitely many smaller bundles and derives formulas for their dimensions using eigenstructure data.

Findings

Closure of a bundle of a matrix pencil is the union of finitely many smaller bundles.

Closure of a bundle of higher-grade matrix polynomials also includes smaller bundles.

Formulas for the dimension of bundles are provided based on eigenstructure characteristics.

Abstract

Bundles of matrix polynomials are sets of matrix polynomials with the same size and grade and the same eigenstructure up to the specific values of the eigenvalues. It is known that the closure of the bundle of a pencil $L$ (namely, a matrix polynomial of grade $1$), denoted by $\mathcal{B}(L)$, is the union of $\mathcal{B}(L)$ itself with a finite number of other bundles. The first main contribution of this paper is to prove that the dimension of each of these bundles is strictly smaller than the dimension of $\mathcal{B}(L)$. The second main contribution is to prove that also the closure of the bundle of a matrix polynomial of grade larger than 1 is the union of the bundle itself with a finite number of other bundles of smaller dimension. To get these results we obtain a formula for the (co)dimension of the bundle of a matrix pencil in terms of the Weyr characteristics of the partial multiplicities of the eigenvalues and of the (left and right) minimal indices, and we provide a characterization for the inclusion relationship between the closures of two bundles of matrix polynomials of the same size and grade.