Families of skew-symmetric matrices of even size

TL;DR

This paper classifies singularities of complex skew-symmetric matrix families of even size, unifying simple classifications of symmetric and arbitrary matrices through suspension and embedding techniques.

Contribution

It introduces a unified approach to classify simple singularities of skew-symmetric, symmetric, and arbitrary matrices using suspension and embedding methods.

Findings

Classification of singularities for skew-symmetric matrices of even size

Embedding of matrix spaces into skew-symmetric matrix spaces

Unified framework for simple matrix family classifications

Abstract

The main result of the paper is a classification of singularities of complex skew-symmetric matrix families of even size which are simple under a natural equivalence relation. The classification is obtained by appropriate suspensions of simple families of arbitrary square matrices classified by Bruce and Tari. The suspensions we are using are based on the embedding of a space of arbitrary square matrices into the space of skew-symmetric matrices of twice the size. We also show that similar relations embed Bruce's classification of simple symmetric matrix families into the Bruce-Tari simple list. Our constructions introduce a unified approach to all three simple classifications.