Matrix representations of octonions and their applications

Abstract

As is well-known, the real quaternion division algebra ℍ is algebraically isomorphic to a 4-by-4 real matrix algebra. But the real division octonion algebra Open image in new window can not be algebraically isomorphic to any matrix algebras over the real number field ℝ, because Open image in new window is a non-associative algebra over ℝ. However since Open image in new window is an extension of 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.