Parametrising correlation matrices

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.