  We study symplectic linear algebra over the ring $\Rt$ of Colombeau
generalized numbers. Due to the algebraic properties of $\Rt$ it is possible to
preserve a number of central results of classical symplectic linear algebra. In
particular, we construct symplectic bases for any symplectic form on a free
$\Rt$-module of finite rank. Further, we consider the general problem of
eigenvalues for matrices over $\Kt$ ($\K=\R$ or $\C$) and derive normal forms
for Hermitian and skew-symmetric matrices. Our investigations are motivated by
applications in non-smooth symplectic geometry and the theory of Fourier
integral operators with non-smooth symbols.
