# Numerical integration for functions of symmetric matrices

This is mostly a reference request. I have integrals of the type $$\int_C f(A) (dA)$$ 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?

• You don't really expect an answer to the general question? Some of these integrals can be quite hard.... Sep 10, 2012 at 8:03
• I really do know that some of these integrals can be quite hard! That is why I look for some references for numerical integration. Sep 10, 2012 at 17:07
• @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. Sep 11, 2012 at 19:36