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.

中文翻译：

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参数化相关矩阵

相关矩阵是正定实数矩阵的子类，对角线上的所有条目都等于 1。早期的工作已经在部分相关性方面以及超球面坐标方面展示了相应 Cholesky 分解的参数化。我们展示了这两者是如何相关的，从 Schur 补集方面的偏相关定义开始。我们将其扩展到将相关矩阵推广到复数和四元数条目的情况。与实际情况一样，我们展示了超球面参数化如何自然地导致相关矩阵 $\{R\}$ 空间上的分布，其概率密度函数与 $( \det R)^a$ 成正比。对于某些 $a$，