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

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

I'm trying to show it using densities with respect to Lebesgue measure. Murihead'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. 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, so I'm working through it at the moment.

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