Symplectic modules over Colombeau-generalized numbers

Abstract

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.