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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?

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  • $\begingroup$ You don't really expect an answer to the general question? Some of these integrals can be quite hard.... $\endgroup$ – Suvrit Sep 10 '12 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 halvorsen Sep 10 '12 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. $\endgroup$ – Suvrit Sep 11 '12 at 19:36
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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.

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