The Exceptional Jordan Eigenvalue Problem

We discuss the eigenvalue problem for 3 ×3 octonionic Hermitian matrices which is relevant to theJordan formulation of quantum mechanics. In contrast tothe eigenvalue problems considered in our previous work, all eigenvalues are real and solve theusual characteristic equation. We give an elementaryconstruction of the corresponding eigenmatrices, and wefurther speculate on a possible application to particle physics.