The Exceptional Jordan Eigenvalue Problem

TL;DR

This paper explores the eigenvalue problem for 3x3 octonionic Hermitian matrices, revealing all eigenvalues are real and providing a construction method, with potential implications for particle physics.

Contribution

It presents a new approach to the eigenvalue problem for octonionic matrices, showing all eigenvalues are real and constructing eigenmatrices explicitly.

Findings

All eigenvalues are real for the considered matrices.

Eigenmatrices can be constructed explicitly.

Potential applications to particle physics are discussed.

Abstract

We discuss the eigenvalue problem for 3x3 octonionic Hermitian matrices which is relevant to the Jordan formulation of quantum mechanics. In contrast to the eigenvalue problems considered in our previous work, all eigenvalues are real and solve the usual characteristic equation. We give an elementary construction of the corresponding eigenmatrices, and we further speculate on a possible application to particle physics.