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Minimal degenerations of orbits of skew-symmetric matrix pencils

Sweta Das, Andrii Dmytryshyn

TL;DR

The work addresses how the eigenstructure of a skew-symmetric pencil $A-\lambda B$ changes under small skew-symmetric perturbations by analyzing closure relations of congruence orbits. It introduces a complete set of four structure-transition rules (and a coin-move interpretation) that yield necessary and sufficient conditions for one orbit to cover another, enabling efficient construction of orbit-stratification graphs. The results also show how to relate stratification graphs across dimensions by adjoining a trivial summand $M_0$, facilitating scalable graph construction. These contributions advance the qualitative understanding of eigenstructure sensitivity in structured perturbations and support practical computation and software implementation for orbit stratifications.

Abstract

Complete eigenstructure, e.g., eigenvalues with multiplicities and minimal indices, of a skew-symmetric matrix pencil may change drastically if the matrix coefficients of the pencil are subjected to (even small) perturbations. These changes can be investigated qualitatively by constructing the stratification (closure hierarchy) graphs of the congruence orbits of the pencils. The results of this paper facilitate the construction of such graphs by providing all closest neighbours for a given node in the graph. More precisely, we prove a necessary and sufficient condition for one congruence orbit of a skew-symmetric matrix pencil, A, to belong to the closure of the congruence orbit of another pencil, B, such that there is no pencil, C, whose orbit contains the closure of the orbit of A and is contained in the closure of the orbit of B.

Minimal degenerations of orbits of skew-symmetric matrix pencils

TL;DR

The work addresses how the eigenstructure of a skew-symmetric pencil changes under small skew-symmetric perturbations by analyzing closure relations of congruence orbits. It introduces a complete set of four structure-transition rules (and a coin-move interpretation) that yield necessary and sufficient conditions for one orbit to cover another, enabling efficient construction of orbit-stratification graphs. The results also show how to relate stratification graphs across dimensions by adjoining a trivial summand , facilitating scalable graph construction. These contributions advance the qualitative understanding of eigenstructure sensitivity in structured perturbations and support practical computation and software implementation for orbit stratifications.

Abstract

Complete eigenstructure, e.g., eigenvalues with multiplicities and minimal indices, of a skew-symmetric matrix pencil may change drastically if the matrix coefficients of the pencil are subjected to (even small) perturbations. These changes can be investigated qualitatively by constructing the stratification (closure hierarchy) graphs of the congruence orbits of the pencils. The results of this paper facilitate the construction of such graphs by providing all closest neighbours for a given node in the graph. More precisely, we prove a necessary and sufficient condition for one congruence orbit of a skew-symmetric matrix pencil, A, to belong to the closure of the congruence orbit of another pencil, B, such that there is no pencil, C, whose orbit contains the closure of the orbit of A and is contained in the closure of the orbit of B.

Paper Structure

This paper contains 7 sections, 14 theorems, 45 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. Matrix pencils under strict equivalence
  4. Skew-symmetric matrix pencils under congruence
  5. Orbit cover relations for skew-symmetric matrix pencils
  6. Relation between closure hierarchy graphs of different dimensions
  7. Result and future work

Key Result

Theorem 2.1

Gant59 Each $m \times n$ matrix pencil $A - \lambda B$ is strictly equivalent to a direct sum, uniquely determined up to permutation of summands, of pencils of the form

Figures (3)

  • Figure 1: Integer partition, $\mathcal{J} = \{4,4,4,4,2\}$, with vertical pair of coins marked as .
  • Figure 2: Orbit stratification for $6\times 6$ skew-symmetric matrix pencils along with the codimensions. Rules from Theorem \ref{['congruence']} used for each structure transition are indicated near the corresponding edge.
  • Figure 3: Orbit stratification for $7\times 7$ skew-symmetric matrix pencil along with the codimensions. Rules from Theorem \ref{['congruence']} used for each structure transition are indicated near the corresponding edge.

Theorems & Definitions (24)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • proof
  • Theorem 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Remark 2.8
  • Lemma 2.9
  • ...and 14 more