Symplectic spaces and pairs of symmetric and nonsingular skew-symmetric matrices under congruence

TL;DR

This paper simplifies the canonical form classification of pairs of symmetric and nonsingular skew-symmetric matrices over fields of characteristic not 2, linking it to quadratic forms and Hamiltonian operators on symplectic spaces.

Contribution

It provides a simpler canonical form for such matrix pairs when the skew-symmetric matrix is nonsingular, enhancing understanding of quadratic forms and Hamiltonian operators.

Findings

Derived a simpler canonical form for pairs with nonsingular skew-symmetric matrices.

Connected matrix pairs to quadratic forms on symplectic spaces.

Applied results to canonical matrices of quadratic forms and Hamiltonian operators.

Abstract

Let $\mathbb F$ be a field of characteristic not $2$, and let $(A,B)$ be a pair of $n\times n$ matrices over $\mathbb F$, in which $A$ is symmetric and $B$ is skew-symmetric. A canonical form of $(A,B)$ with respect to congruence transformations $(S^TAS,S^TBS)$ was given by Sergeichuk (1988) up to classification of symmetric and Hermitian forms over finite extensions of $\mathbb F$. We obtain a simpler canonical form of $(A,B)$ if $B$ is nonsingular. Such a pair $(A,B)$ defines a quadratic form on a symplectic space, that is, on a vector space with scalar product given by a nonsingular skew-symmetric form. As an application, we obtain known canonical matrices of quadratic forms and Hamiltonian operators on real and complex symplectic spaces.