On equivalence of the LKJ distribution and the restricted Wishart distribution

In this paper, we want to show the Restricted Wishart distribution is equivalent to the LKJ distribution, which is one way to specify a uniform distribution from the space of positive definite correlation matrices. Based on this theorem, we propose a new method to generate random correlation matrices from the LKJ distribution. This new method is faster than the original onion method for generating random matrices, especially in the low dimension ($T<120$) situation.