Symplectic matrices with predetermined left eigenvalues

TL;DR

This paper proves that for any four unit quaternion numbers, a corresponding 2x2 symplectic matrix exists with those as left eigenvalues, and demonstrates an application to Lie group topology.

Contribution

It provides a constructive proof that any four unit quaternions can be realized as left eigenvalues of a 2x2 symplectic matrix, linking quaternion eigenvalues to symplectic matrices.

Findings

Existence of symplectic matrices with prescribed left eigenvalues

Constructive method for finding such matrices

Application to LS category of Lie groups

Abstract

We prove that given four arbitrary quaternion numbers of norm 1 there always exists a $2\times 2$ symplectic matrix for which those numbers are left eigenvalues. The proof is constructive. An application to the LS category of Lie groups is given.