The title says most. Let $P\_p$ be the cone of positive-definite $p \times p$ matrices. 


One can define the Laplace transform of (the distribution of) a random matrix with values in $P\_p$ by (for instance, Muirhead: "Aspects of Multivariate Analysis") the integral


\begin{equation}
 \phi(\Theta) = \int\_{P\_p} \exp(\sum\_{j\le k}^p \theta\_{jk} a\_{jk}) f(A)\; dA
\end{equation}


where $f(A)$ is the density function of $A$. (And $\Theta$ is a symmetric $p\times p$-matrix)


So the question is: I am searching for references for this Laplace transform, inversion theorems, numerical methods, known transform formulas, .... etc ???


Thanks for answers! Now, I am searching for those references, but one is really difficult to find, that is, volume 2 of Audrey Terras' book: "Harmonic analysis on symmetric spaces II"


I have found the first volume, but volume II cannot even be found on Springers own website! Any ideas about how to find it?



Laplace transform on the cone of positive-definite matrices
Laplace and Fourier transforms on the cone of positive definite matrices are somewhat well-studied. The framework is, as already mentioned by Alexander, that of "analysis on symmetric spaces." Here are some references where this subject is developed in greater detail:


1. *Harmonic analysis on symmetric spaces and applications*, vols. I and II (particularly chapter 4 for vol. II covers positive matrices), by A. Terras, Springer Verlag. I think Terras also lists a short table of Fourier transforms and their inverses; those should help with Laplace transforms too.
2. *Analysis on symmetric cones*, by J. Faraut, A. Koranyi, Clarendon Press, 1994. In particular, have a look at Chapter VII
3. *Geometric analysis on symmetric spaces*, S. Helgason, AMS, 1991.(Chapter III discusses the Fourier transform)


For computation, it seems you have to do eigenvector decomposition, and numerical integration thereafter. On a related note, I mention the following toolbox by P. Koev:


**[Hypergeometric function of a matrix argument](http://www-math.mit.edu/~plamen/software/mhgref.html)**


I mention this because these functions are closely related to the Laplace transform that you have mentioned (as detailed out in Muirhead's book).


