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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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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