Diagonal Plus Tridiagonal Representatives for Symplectic Congruence Classes of Symmetric Matrices

Đoković, D. Ž. ; Szechtman, F. ; Zhao, K.

Canadian mathematical bulletin, Tome 48 (2005) no. 3, pp. 394-404

Résumé

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.

Détail

DOI : 10.4153/CMB-2005-036-x

Mots-clés : 11E39, 15A63, 17B20