# Showing a useful result for Wisharts and Multivariate Beta random matrices

Let $$\mathbf{A} \sim \text{Wishart}_m\left(k_a,\mathbf{V} \right)$$ and $$\mathbf{B} \sim \text{Wishart}_m\left(k_b,\mathbf{V} \right)$$ be two full rank Wishart random matrices. Define $$\mathbf{S} = \mathbf{A} + \mathbf{B}$$ and $$\mathbf{U} = (\mathbf{T}^{-1})^{'}\mathbf{A}\mathbf{T}^{-1}$$ where $$\mathbf{T}'\mathbf{T}$$ is the Cholesky decomposition of $$\mathbf{S}$$. Show

1. $$\mathbf{S} \sim \text{Wishart}\left(k_1 + k_2, \mathbf{V} \right)$$
2. $$\mathbf{U} \sim \text{Matrix Beta}_m\left(\frac{k_1}{2}, \frac{k_2}{2}\right)$$, and
3. $$\mathbf{S}$$ is independent of $$\mathbf{U}$$.

I'm trying to show it using densities with respect to Lebesgue measure. Muirhead's book goes through a lot of this stuff, but appeals to k-forms, which I'm not very comfortable with, and I'm trying to avoid at the moment. Actually, I think this book even defines the matrix-variate beta distribution using this. Apparently it's also true in the not-full-rank case (c.f. Uhlig 1994), but I'd like to tackle the simpler version first.

This book looks pretty good, so I'm working through it at the moment.