Matrix representations of octonions and their applications

  • Yongge TianEmail author


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
can not be algebraically isomorphic to any matrix algebras over the real number field ℝ, because
is a non-associative algebra over ℝ. However since
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.

AMS Mathematics Subject Classification

15A33 15A06 15A24 17A34 

Key words

quaternions octonions matrix representations linear equations similarity eigenvalues Cayley-Hamilton theorem 


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Copyright information

© Birkhäuser-Verlag AG 2000

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsQueen’s UniversityKingstonCanada

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