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

Abstract

Let F be a field of characteristic not 2, and let (A,B) be a pair of n×n matrices over F, in which A is symmetric and B is skew-symmetric. A canonical form of (A,B) with respect to congruence transformations (STAS,STBS) was given by Sergeichuk (1988) [25] up to classification of symmetric and Hermitian forms over finite extensions of 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.