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On the dimension of orbits of matrix pencils under strict equivalence

Fernando De Terán, Froilán M. Dopico, Patryk Pagacz

TL;DR

The paper addresses how the dimension of the orbit of a matrix pencil under strict similarity behaves under orbit-closure containment. It provides a direct, inequality-based proof that if $M\in\overline{{\cal O}(L)}$, then $\dim{\cal O}(M)\le\dim{\cal O}(L)$, with equality precisely when $M\in{\cal O}(L)$; this hinges on majorization criteria for the Weyr characteristics and a codimension formula expressed through the Kronecker canonical form data. The authors leverage six elementary transformations that realize orbit-closure inclusions and carefully compare the contributions of finite and infinite eigenvalues, Jordan blocks, and singular blocks. The result clarifies the geometry of pencil orbits and provides a concrete condition for when an orbit is strictly larger than another, with potential impact on perturbation analysis and eigenstructure-based applications.

Abstract

We prove that, given two matrix pencils $L$ and $M$, if $M$ belongs to the closure of the orbit of $L$ under strict equivalence, then the dimension of the orbit of $M$ is smaller than or equal to the dimension of the orbit of $L$, and the equality is only attained when $M$ belongs to the orbit of $L$. Our proof uses only the majorization involving the eigenstructures of $L$ and $M$ which characterizes the inclusion relationship between orbit closures, together with the formula for the codimension of the orbit of a pencil in terms of its eigenstruture.

On the dimension of orbits of matrix pencils under strict equivalence

TL;DR

The paper addresses how the dimension of the orbit of a matrix pencil under strict similarity behaves under orbit-closure containment. It provides a direct, inequality-based proof that if , then , with equality precisely when ; this hinges on majorization criteria for the Weyr characteristics and a codimension formula expressed through the Kronecker canonical form data. The authors leverage six elementary transformations that realize orbit-closure inclusions and carefully compare the contributions of finite and infinite eigenvalues, Jordan blocks, and singular blocks. The result clarifies the geometry of pencil orbits and provides a concrete condition for when an orbit is strictly larger than another, with potential impact on perturbation analysis and eigenstructure-based applications.

Abstract

We prove that, given two matrix pencils and , if belongs to the closure of the orbit of under strict equivalence, then the dimension of the orbit of is smaller than or equal to the dimension of the orbit of , and the equality is only attained when belongs to the orbit of . Our proof uses only the majorization involving the eigenstructures of and which characterizes the inclusion relationship between orbit closures, together with the formula for the codimension of the orbit of a pencil in terms of its eigenstruture.
Paper Structure (3 sections, 5 theorems, 23 equations)

This paper contains 3 sections, 5 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Basic notions and results
  3. The main result

Key Result

Theorem 1

If $L$ and $M$ are matrix pencils of the same size and $h:={\rm rank\,} L-{\rm rank\,} M\geq0$, then $M\in\overline{\cal O}(L)$ if and only if the following three majorizations hold: (1) $r(M) \prec_{ \rm w} r(L)+(h,h,\ldots)$; (2) $\ell(M) \prec_{ \rm w} \ell(L)+(h,h,\ldots)$; and (3) $W(\lambda,L)

Theorems & Definitions (7)

  • Theorem 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Theorem 4
  • Theorem 5