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.
Categories: math.RA