  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 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 $X^\dagger X$, when $X$ has random octonion entries.
Analytic results are obtained from $N=2$, but are observed to break down in the
$3 \times 3$ case.
