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

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 $(S^{T}AS,S^{T}BS)$ 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.