Left eigenvalues of $2\times 2$ symplectic matrices

TL;DR

This paper characterizes 2x2 symplectic matrices with infinitely many left eigenvalues, providing new proofs and applying an algorithm for solving quaternionic matrix equations.

Contribution

It offers a complete characterization of such matrices and introduces a novel proof approach using an existing algorithm.

Findings

Identifies conditions for infinite left eigenvalues in 2x2 symplectic matrices

Provides a new proof of a known quaternionic eigenvalue result

Applies an algorithm for solving quaternionic matrix equations

Abstract

We obtain a complete characterization of the $2\times 2$ symplectic matrices having an infinite number of left eigenvalues. Previously, we give a new proof of a result from Huang and So about the number of eigenvalues of a quaternionic matrix. This is achieved by applying an algorithm for the resolution of equations due to De Leo et al.