Finding octonionic eigenvectors using Mathematica Original Research Article

The eigenvalue problem for 3 × 3 octonionic Hermitian matrices contains some surprises, which we have reported elsewhere [T. Dray, C.A. Manogue, The Octonionic Eigenvalue Problem, math.RA/9807126, Adv. Appl. Clifford Algebras, in press]. In particular, the eigenvalues need not be real, there are 6 rather than 3 real eigenvalues, and the corresponding eigenvectors are not orthogonal in the usual sense. The nonassociativity of the octonions makes computations tricky, and all of these results were first obtained via brute force (but exact) Mathematica computations. Some of them, such as the computation of real eigenvalues, have subsequently been implemented more elegantly; others have not. We describe here the use of Mathematica in analyzing this problem, and in particular its use in proving a generalized orthogonality property for which no other proof is known.