Minimal degenerations of orbits of skew-symmetric matrix pencils

TL;DR

This paper studies how the eigenstructure of skew-symmetric matrix pencils changes under small perturbations by characterizing minimal degenerations of their orbits, aiding the construction of stratification graphs.

Contribution

It provides necessary and sufficient conditions for orbit closures of skew-symmetric matrix pencils, enabling precise understanding of their degenerations and orbit adjacency.

Findings

Characterization of minimal orbit degenerations

Necessary and sufficient conditions for orbit closure inclusion

Facilitation of stratification graph construction

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.