  Correlation matrices are the sub-class of positive definite real matrices
with all entries on the diagonal equal to unity. Earlier work has exhibited a
parametrisation of the corresponding Cholesky factorisation in terms of partial
correlations, and also in terms of hyperspherical co-ordinates. We show how the
two are relating, starting from the definition of the partial correlations in
terms of the Schur complement. We extend this to the generalisation of
correlation matrices to the cases of complex and quaternion entries. As in the
real case, we show how the hyperspherical parametrisation leads naturally to a
distribution on the space of correlation matrices $\{R\}$ with probability
density function proportional to $( \det R)^a$. For certain $a$, a construction
of random correlation matrices realising this distribution is given in terms of
rectangular standard Gaussian matrices.
