Finding 3x3 Hermitian Matrices over the Octonions with Imaginary Eigenvalues

TL;DR

This paper explores the spectral properties of 3x3 Hermitian matrices over octonions, revealing that certain imaginary octonionic vectors are eigenvectors with eigenvalues linked to their associator, expanding understanding of octonionic linear algebra.

Contribution

It demonstrates that specific imaginary octonionic vectors are eigenvectors of a family of Hermitian matrices with eigenvalues related to their associator, a novel insight into octonionic eigenvalue problems.

Findings

Imaginary octonionic vectors are eigenvectors of Hermitian matrices.

Eigenvalues are equal to the associator of the vector's components.

Identifies a 6-parameter family of such matrices.

Abstract

We show that any 3-component octonionic vector which is purely imaginary, but not quaternionic, is an eigenvector of a 6-parameter family of Hermitian octonionic matrices, with imaginary eigenvalue equal to the associator of its elements.