On a Quaternionic Representation for Sp(4, R)

TL;DR

This paper introduces a quaternionic representation for 4x4 real symplectic matrices, enabling easier computation of polar decompositions and characterization of positive definite matrices without spectral analysis.

Contribution

It develops a novel quaternionic formula for symplectic matrices in dimension four, simplifying polar decomposition and matrix characterization processes.

Findings

Provides explicit quaternionic formulas for symplectic matrices

Enables polar decomposition via 2x2 linear system solutions

Characterizes symplectic positive definite matrices in dimension four

Abstract

This work provides a quaternioinc reprsentation for real symplectic matrices in dimension four, analogous to the pair of unit quaternions representation for special orthogonal matrices. In the process of finding formulae for this representation in terms of the entries of the symplectic matrix being thereby represented, it shows how to compute the polar decomposition of a symplectic matrix without any need for spectral calculation. The technique just requires the solution of a simple 2x2 linear system. The work also characterizes symplectic, positive definite matrices in dimension four, and thus can be used in applications where the non-compact part of the symplectic group in dimension four is of utility.