Diagonal Plus Tridiagonal Representatives for Symplectic Congruence Classes of Symmetric Matrices

[摘要] Let $n=2m$ be even and denote by $Sp_n(F)$ the symplectic groupof rank $m$ over an infinite field $F$ of characteristic differentfrom $2$. We show that any $nimes n$ symmetric matrix $A$ isequivalent under symplectic congruence transformations to thedirect sum of $mimes m$ matrices $B$ and $C$, with $B$ diagonaland $C$ tridiagonal. Since the $Sp_n(F)$-module of symmetric$nimes n$ 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.

[学科分类] 数学（综合）