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Finding octonionic eigenvectors using Mathematica. (English) Zbl 1002.65046

Summary: The eigenvalue problem for \(3\times 3\) octonionic Hermitian matrices contains some surprises, which we have reported elsewhere [cf. T. Dray and C. A. Manogue, Adv. Appl. Clifford Algebr. 8, No. 2, 341-365 (1998; Zbl 0927.15007)]. 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.

MSC:

65F15 Numerical computation of eigenvalues and eigenvectors of matrices
15B33 Matrices over special rings (quaternions, finite fields, etc.)
15A18 Eigenvalues, singular values, and eigenvectors

Citations:

Zbl 0927.15007
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References:

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