TY - JOUR

T1 - Symplectic spaces and pairs of symmetric and nonsingular skew-symmetric matrices under congruence

AU - Bovdi, Victor A.

AU - Horn, Roger A.

AU - Salim, Mohamed A.

AU - Sergeichuk, Vladimir V.

N1 - Funding Information:
The work was supported in part by the UAEU UPAR grants G00001922 and G00002160 .
Publisher Copyright:
© 2017 Elsevier Inc.

PY - 2018/1/15

Y1 - 2018/1/15

N2 - 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.

AB - 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.

KW - Hamiltonian operators

KW - Pairs of symmetric and skew-symmetric matrices

KW - Symplectic congruence

KW - Symplectic spaces

UR - http://www.scopus.com/inward/record.url?scp=85030152308&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85030152308&partnerID=8YFLogxK

U2 - 10.1016/j.laa.2017.09.026

DO - 10.1016/j.laa.2017.09.026

M3 - Article

AN - SCOPUS:85030152308

SN - 0024-3795

VL - 537

SP - 84

EP - 99

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

ER -