In a paper by Mathai, he uses the following 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 $0\lt U \lt I$ (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?



Integral representation of a determinant
*OLDER EDIT. (Elementary derivation)* I realized that my original answer was actually *overkill* for the question. The said integral in question follows from the definition of the *multivariate Gamma function*


\begin{equation\*}
\Gamma\_p(a) := \int\_{A > 0} \exp(-\mbox{tr}(A))\det(A)^{a-(p+1)/2}(dA),
\end{equation\*}
where $\Re(a)>(p-1)/2$. 


From this it follows (by a change of variables) that for a positive definite matrix $S$, 
\begin{equation\*}
\int\_{A > 0} \exp(-\mbox{tr}(S^{-1}A))\det(A)^{a-(p+1)/2}(dA) = \Gamma\_p(a)\det(S)^a,
\end{equation\*}
so that with $S=(I-U)^{-1}$ we obtain the integral in question. 


Of course, to complete the picture it may be helpful to express $\Gamma\_p(a)$ in more elementary terms. Chapter 2 of Muirhead's book provides these details. I cite the result that provides this expression.



> 
> **Theorem (Muirhead (1982), Thm 2.1.2)** Let $\Re(a) > (p-1)/2$. Then,
>  $$
> \Gamma\_p(a) = \pi^{p(p-1)/4}\prod\_{j=1}^p \Gamma(a - (j-1)/2)
> $$
> 
> 
> 


(Hint: To prove the above, write the *Cholesky decomposition* $A=T'T$, with these change of variables, the original Gamma integral factorizes into a product of Gaussian and Gamma integrals.)




---


The part that I recalled below provides yet another representation that expresses the multiplicative determinantal lhs in terms of an infinite sum.


**OLDER STUFF**


This is actually somewhat classical knowledge. Here are two related pointers.


A *partition* $\tau=(t\_1,\ldots,t\_m)$ is a vector of nonnegative integers listed in increasing order, and $|\tau|$ denotes $t\_1+\cdots+t\_m$. The *generalized Pochhammer* symbol $(a)\_\tau$ is defined as
\begin{equation\*}
\newcommand{\risingf}[2]{{{#1}}^{\overline{{#2}}}}
 (a)\_\tau := \frac{\Gamma\_d(a+\tau)}{\Gamma\_d(a)} = \prod\_{l=1}^m \risingf{\bigl(a - \tfrac{1}{2}(l-1)\bigr)}{t\_l}
\end{equation\*}


Let $C\_\tau(X)$ be the *Zonal Polynomial* with signature partition $\tau$. Then, the following representation exists


For a matrix $U$ satisfying $\|U\| < 1$, we have the following "binomial-theorem"


\begin{equation}
 \frac{1}{|I-U|^a} = \sum\_{k\ge 0}\sum\_{|\tau| = k} \frac{(a)\_\tau C\_\tau(U)}{k!}.
\end{equation}


Using representations for these Zonal polynomials, one can obtain the integral representation mentioned in the original post.


More directly, you can look at Chapter 7 of *R. Muirhead, "Aspects of Multivariate Statistical Theory"*, where you'll see that actually, $|I-U|^{-a}={}\_1F\_0(a;U)$, a matrix argument hypergeometric function. I've to run now, if I get a chance I'll clean up my answer and fill in the details.


