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Let $\mathbb F$ be a field of characteristic not $2$, and let $(A,B)$ be apair 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 congruencetransformations $(S^TAS,S^TBS)$ was given by Sergeichuk (1988) up toclassification of symmetric and Hermitian forms over finite extensions of$\mathbb F$. We obtain a simpler canonical form of $(A,B)$ if $B$ isnonsingular. Such a pair $(A,B)$ defines a quadratic form on a symplecticspace, that is, on a vector space with scalar product given by a nonsingularskew-symmetric form. As an application, we obtain known canonical matrices ofquadratic forms and Hamiltonian operators on real and complex symplecticspaces.

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