Normal Forms for Symplectic Matrices

TL;DR

This paper provides a straightforward, geometrically motivated method to derive normal forms for symplectic matrices, expressing them through elementary Jordan matrices and signature-related integers.

Contribution

It introduces a self-contained, elementary approach to symplectic matrix normal forms based on geometric considerations and quadratic form signatures.

Findings

Normal forms expressed via elementary Jordan matrices

Use of signatures of quadratic forms in classification

Elementary, geometrically motivated derivation

Abstract

We give a self contained and elementary description of normal forms for symplectic matrices, based on geometrical considerations. The normal forms in question are expressed in terms of elementary Jordan matrices and integers with values in $\{-1,0,1\}$ related to signatures of quadratic forms naturally associated to the symplectic matrix.