In a paper by Mathai, he uses the following integrap integral representation of a determinant, (or, really, what I give is a simple special case of what he gives), without any explication. All matrices are real $p\times p$ symmetric positive definite.
\begin{equation} | I-U |^{-a} = \frac{1}{\Gamma_p(a)} \int_{T>0} |T|^{a-(p+1)/2} \exp(-\text{Tr}(I-U)T) \;dT \end{equation}
where $U$ satisfies $0(p-1)/2$.0\lt U \lt I$ (in the cone order in the cone of positive definte matrices), the integral is over the cone of positive definite matrices and $\Gamma_p(a)$ is the generalized gamma function in dimension $p$, and $\Re(a) > (p-1)/2$.
Any references for this?
