Matrix Representations of Octonions and Their Applications

Abstract

As is well-known, the real quaternion division algebra $\mathcal{H}$ is algebraically isomorphic to a 4-by-4 real matrix algebra. But the real division octonion algebra $\mathcal{O}$ can not be algebraically isomorphic to any matrix algebras over the real number field $\mathcal{R}$, because $\mathcal{O}$ is a non-associative algebra over $\mathcal{R}$. However since $\mathcal{O}$ is an extension of $\mathcal{H}$ 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.

23 pages, LaTex