This is mostly a reference request. I have integrals of the type
\begin{equation}
 \int\_C f(A) (dA)
\end{equation}
where $f$ is a real-valued function of a positive-(semi)definite matrix argument, and the integration region $C$ is an interval in the cone of positive-definite matrices, such as
$C=[0.I]$, where this cone interval denotes the set of all positive-definite matrices with positive eigenvalues all less than one. Other cone intervals could also occur, but in most cases they can be transformed to this or a similar form. Mostly the unctions $f$ will be symmetric functions in the sense that $f(AB)=f(BA)$, where $A$ and $B$ are positive-definite matrices.There must be some papers about this kind of problem?



Numerical integration for functions of symmetric matrices
I think that you could transform the problem through the use of tools from Chapter 5 in the following book


* [Eaton (2007). Multivariate Statistics: A Vector Space Approach](http://projecteuclid.org/euclid.lnms/1196285102)


There are explicit examples there for integration over spaces of symmetric positive definite matrices.


