Reduction of a pair of skew-symmetric matrices to its canonical form under congruence

TL;DR

This paper presents an algorithm for reducing pairs of skew-symmetric matrices to their canonical form under congruence, including a constructive proof of the canonical form over algebraically closed fields.

Contribution

It introduces a new algorithm for regularization decomposition of skew-symmetric matrix pairs and provides a constructive proof of their canonical form under congruence.

Findings

Algorithm for regularization decomposition of skew-symmetric pairs

Constructive proof of canonical form under congruence

Applicable over algebraically closed fields of characteristic not 2

Abstract

Let $(A,B)$ be a pair of skew-symmetric matrices over a field of characteristic not 2. Its regularization decomposition is a direct sum \[ (\underline{\underline A},\underline{\underline B})\oplus (A_1,B_1)\oplus\dots\oplus(A_t,B_t) \] that is congruent to $(A,B)$, in which $(\underline{\underline A},\underline{\underline B})$ is a pair of nonsingular matrices and $(A_1,B_1),$ $\dots,$ $(A_t,B_t)$ are singular indecomposable canonical pairs of skew-symmetric matrices under congruence. We give an algorithm that constructs a regularization decomposition. We also give a constructive proof of the known canonical form of $(A,B)$ under congruence over an algebraically closed field of characteristic not 2.