  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.
