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Diagonal Plus Tridiagonal Representatives for Symplectic Congruence Classes of Symmetric Matrices

Published online by Cambridge University Press:  20 November 2018

D. Ž. Đoković ,
F. Szechtman  and
K. Zhao
D. Ž. Đoković
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, ON, N2L 3G1 e-mail: djokovic@uwaterloo.ca
F. Szechtman
Affiliation:
Department of Mathematics and Statistics, University of Regina, Regina, SK, S4S 0A2 e-mail: szechtf@math.uregina.ca
K. Zhao
Affiliation:
Department of Mathematics, Wilfrid Laurier University, Waterloo, ON, N2L 3C5 and Mathematics Department, Henan University, Henan, China e-mail: kzhao@wlu.ca
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Abstract

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Let $n=2m$ be even and denote by $\text{S}{{\text{p}}_{n}}\left( F \right)$ the symplectic group of rank $m$ over an infinite field $F$ of characteristic different from 2. We show that any $n\times n$ symmetric matrix $A$ is equivalent under symplectic congruence transformations to the direct sum of $m\times m$ matrices $B$ and $C$, with $B$ diagonal and $C$ tridiagonal. Since the $\text{S}{{\text{p}}_{n}}\left( F \right)$-module of symmetric $n\times n$ matrices over $F$ is isomorphic to the adjoint module $\mathfrak{s}{{\mathfrak{p}}_{n}}\left( F \right)$, we infer that any adjoint orbit of $\text{S}{{\text{p}}_{n}}\left( F \right)$ in $\mathfrak{s}{{\mathfrak{p}}_{n}}\left( F \right)$ has a representative in the sum of $3m-1$ root spaces, which we explicitly determine.

Keywords

11E3915A6317B20
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 48 , Issue 3 , 01 September 2005 , pp. 394 - 404
Copyright
Copyright © Canadian Mathematical Society 2005

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