Finding Octonionic Eigenvectors Using Mathematica

Details Finding Octonionic Eigenvectors Using Mathematica Finding Octonionic Eigenvectors Using Mathematica

1998-07-23

arxiv.org/abs/math/9807133v1

Comput. Phys. Comm. 115, 536-547 (1998)

Mathematics

Rings and Algebras

LaTeX2e, 22 pages, 8 PS figures (uses included PS prolog; needs elsart.cls and one of epsffig, epsf, graphicx)

Scientific paper

10.1016/S0010-4655(98)00126-X

The eigenvalue problem for 3x3 octonionic Hermitian matrices contains some surprises, which we have reported elsewhere. 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.

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