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# Diagonal Plus Tridiagonal Representatives for Symplectic Congruence Classes of Symmetric Matrices

Published:2005-09-01
Printed: Sep 2005
• D. Ž. Đoković
• F. Szechtman
• K. Zhao
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## Abstract

Let $n=2m$ be even and denote by $\Sp_n(F)$ 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 $\Sp_n(F)$-module of symmetric $n\times n$ matrices over $F$ is isomorphic to the adjoint module $\sp_n(F)$, we infer that any adjoint orbit of $\Sp_n(F)$ in $\sp_n(F)$ has a representative in the sum of $3m-1$ root spaces, which we explicitly determine.
 MSC Classifications: 11E39 - Bilinear and Hermitian forms 15A63 - Quadratic and bilinear forms, inner products [See mainly 11Exx] 17B20 - Simple, semisimple, reductive (super)algebras