Resumen de Diagonal Plus Tridiagonal Representatives for Symplectic Congruence Classes of Symmetric Matrices

D.Z. Djokovíc, Fernando Szechtman, K. Zhao

• Let n = 2m be even and denote by Spn(F) the symplectic group of rank m over an infinite field F of characteristic different from 2. We show that any n x n symmetric matrix A is equivalent under symplectic congruence transformations to the direct sum of m x m matrices B and C, with B diagonal and C tridiagonal. Since the Spn(F)-module of symmetric n x n matrices over F is isomorphic to the adjoint module spn(F), we infer that any adjoint orbit of Spn(F) in spn(F) has a representative in the sum of 3m - 1 root spaces, which we explicitly determine.