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

Original language | English |
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Pages (from-to) | 84-99 |

Number of pages | 16 |

Journal | Linear Algebra and Its Applications |

Volume | 537 |

DOIs | |

Publication status | Published - Jan 15 2018 |

## Keywords

- Hamiltonian operators
- Pairs of symmetric and skew-symmetric matrices
- Symplectic congruence
- Symplectic spaces

## ASJC Scopus subject areas

- Algebra and Number Theory
- Numerical Analysis
- Geometry and Topology
- Discrete Mathematics and Combinatorics