The octonions are one of the four normed division algebras, together with the real, complex and quaternion number systems. The latter three hold a primary place in random matrix theory, where in applications to quantum physics they are determined as the entries of ensembles of Hermitian random matrices by symmetry considerations. Only for N=2 is there an existing analytic theory of Hermitian random matrices with octonion entries. We use a Jordan algebra viewpoint to provide an analytic theory for N=3. We then proceed to consider the matrix structure X†X, when X has random octonion entries. Analytic results are obtained from N=2, but are observed to break down in the 3×3 case.
- Received October 25, 2016.
- Accepted March 3, 2017.
- © 2017 The Author(s)
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