Miniversal deformations of pairs of skew-symmetric matrices under congruence

TL;DR

This paper constructs minimal-parameter normal forms for pairs of skew-symmetric matrices under congruence, enabling analysis of small perturbations and their effects on eigenvalues and indices.

Contribution

It introduces explicit miniversal deformations for skew-symmetric matrix pairs under congruence, with bounds on perturbation distances and applications to stability analysis.

Findings

Normal forms with minimal parameters are derived.

An upper bound on deformation distance is established.

Application to analyzing eigenvalue and index changes under perturbations.

Abstract

Miniversal deformations for pairs of skew-symmetric matrices under congruence are constructed. To be precise, for each such a pair $(A,B)$ we provide a normal form with a minimal number of independent parameters to which all pairs of skew-symmetric matrices $(\widetilde{A},\widetilde{B})$, close to $(A,B)$ can be reduced by congruence transformation which smoothly depends on the entries of the matrices in the pair $(\widetilde{A},\widetilde{B})$. An upper bound on the distance from such a miniversal deformation to $(A,B)$ is derived too. We also present an example of using miniversal deformations for analyzing changes in the canonical structure information (i.e. eigenvalues and minimal indices) of skew-symmetric matrix pairs under perturbations.