Abstract
We study symplectic linear algebra over the ring of Colombeau generalized numbers. Due to the algebraic properties of
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
-module of finite rank. Further, we consider the general problem of eigenvalues for matrices over
(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.
ACKNOWLEDGMENT
We thank an anonymous referee for several suggestions that have substantially improved the presentation.
Notes
For example, c = e
S
with S ⊆ I such that and
would do.
If μ(1 − 2r) = 0, then μ = 2rμ, and hence μr = 2μr 2 = 2μr, i.e., μr = 0 and thus μ = 0.
Communicated by I. Shestakov.