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On bundle closures of matrix pencils and matrix polynomials

Fernando De Terán, Froilán M. Dopico, Vadym Koval, Patryk Pagacz

TL;DR

The paper addresses how the eigenstructure-based bundles of matrix pencils and matrix polynomials sit inside their closures under perturbations. It develops a Weyr-characteristics framework to derive a codimension formula for pencil orbits and proves that the closure of any bundle is a finite union of the bundle with strictly lower-dimensional bundles, both for grade-1 pencils and for grade-$d>1$ polynomials. It also provides a detailed characterization of when one bundle lies in the closure of another via coalescence of eigenvalues and a correspondence with companion pencils, enabling a unified description across grades. The results deepen the geometric understanding of spectral perturbations and have implications for numerical eigenstructure computation, while leaving open extensions to structured cases and further simplifications via Weyr data.

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.

On bundle closures of matrix pencils and matrix polynomials

TL;DR

The paper addresses how the eigenstructure-based bundles of matrix pencils and matrix polynomials sit inside their closures under perturbations. It develops a Weyr-characteristics framework to derive a codimension formula for pencil orbits and proves that the closure of any bundle is a finite union of the bundle with strictly lower-dimensional bundles, both for grade-1 pencils and for grade- polynomials. It also provides a detailed characterization of when one bundle lies in the closure of another via coalescence of eigenvalues and a correspondence with companion pencils, enabling a unified description across grades. The results deepen the geometric understanding of spectral perturbations and have implications for numerical eigenstructure computation, while leaving open extensions to structured cases and further simplifications via Weyr data.

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 (namely, a matrix polynomial of grade ), denoted by , is the union of 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 . 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.
Paper Structure (7 sections, 10 theorems, 53 equations, 2 figures, 1 table)

This paper contains 7 sections, 10 theorems, 53 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Basic notions and notation
  3. Formula for the codimension of the orbit in terms of Weyr characteristics
  4. Main results
  5. Dimension inequalities for bundles of matrix pencils
  6. Bundle closures of matrix polynomials of grade larger than $1$
  7. Conclusions and open questions

Key Result

lemma 1

Let $P(\lambda)$ be an $m\times n$ matrix polynomial with grade $d$. Then

Figures (2)

  • Figure 1: Ferrers diagram for an eigenvalue $\mu$ with partial multiplicities $4$, $3$, $3$, $3$, and $1$, so $q(\mu)=(4,3,3,3,1)$ and $W(\mu)=(5,4,4,1)$.
  • Figure 2: Ferrers diagram for right minimal indices $4$, $3$, $3$, $3$, and $1$, including the zero row, so $a=(4,3,3,3,1)$ and $r=(5,5,4,4,1)$ and weights obtained in the last column.

Theorems & Definitions (20)

  • definition 1
  • lemma 1
  • proof
  • lemma 2
  • proof
  • proposition 1
  • proof
  • theorem 1
  • lemma 3
  • proof
  • ...and 10 more