A topological approach to left eigenvalues of quaternionic matrices

TL;DR

This paper introduces a topological method to analyze left eigenvalues of quaternionic matrices, extending the approach from 2x2 to 3x3 matrices, addressing classification challenges.

Contribution

It develops a topological framework for understanding quaternionic eigenvalues and extends it to higher-order matrices, overcoming limitations of algebraic proofs.

Findings

Topological degree computation for 2x2 matrices

Extension of techniques to 3x3 matrices

Insights into the classification of quaternionic eigenvalues

Abstract

It is known that a $2\times 2$ quaternionic matrix has one, two or an infinite number of left eigenvalues, but the available algebraic proofs are difficult to generalize to higher orders. In this paper a different point of view is adopted by computing the topological degree of a characteristic map associated to the matrix and discussing the rank of the differential. The same techniques are extended to $3\times 3$ matrices, which are still lacking a complete classification.