Matrix Representations of Octonions and Their Applications

Details Matrix Representations of Octonions and Their Applications Matrix Representations of Octonions and Their Applications

2000-03-26

arxiv.org/abs/math/0003166v2

Mathematics

Rings and Algebras

23 pages, LaTex

Scientific paper

As is well-known, the real quaternion division algebra $ {\cal H}$ is algebraically isomorphic to a 4-by-4 real matrix algebra. But the real division octonion algebra ${\cal O}$ can not be algebraically isomorphic to any matrix algebras over the real number field ${\cal R}$, because ${\cal O}$ is a non-associative algebra over ${\cal R}$. However since ${\cal O}$ is an extension of ${\cal H}$ by the Cayley-Dickson process and is also finite-dimensional, some pseudo real matrix representations of octonions can still be introduced through real matrix representations of quaternions. In this paper we give a complete investigation to real matrix representations of octonions, and consider their various applications to octonions as well as matrices of octonions.

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