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 realvalued function of a positive(semi)definite matrix argument, and the integration region $C$ is an interval in the cone of positivedefinite matrices, such as $C=[0.I]$, where this cone interval denotes the set of all positivedefinite 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 positivedefinite matrices.There must be some papers about this kind of problem?

$\begingroup$ You don't really expect an answer to the general question? Some of these integrals can be quite hard.... $\endgroup$– SuvritSep 10, 2012 at 8:03

$\begingroup$ I really do know that some of these integrals can be quite hard! That is why I look for some references for numerical integration. $\endgroup$– kjetil b halvorsenSep 10, 2012 at 17:07

$\begingroup$ @Kjetil: I meant that even numerically they can be quite hard. But actually, if you write $A=UDU^*$, and $f$ is nice (e.g., unitarily invariant), you should be able to get many of these integrals into a more practical form. $\endgroup$– SuvritSep 11, 2012 at 19:36
1 Answer
I think that you could transform the problem through the use of tools from Chapter 5 in the following book
There are explicit examples there for integration over spaces of symmetric positive definite matrices.