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

Victor A. Bovdi, Roger A. Horn, Mohamed A. Salim, Vladimir V. Sergeichuk

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)

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.

Original languageEnglish
Pages (from-to)84-99
Number of pages16
JournalLinear Algebra and Its Applications
Volume537
DOIs
Publication statusPublished - 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

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