# Laplace transform on the cone of positive-definite matrices

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 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

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

Second editions of both volumes are about to be published. It will probably take a year to update them though. audrey

• Oh that is so wonderful to hear; it was with such great difficulty that I managed to lay hands on your books. Looking forward! – Suvrit Apr 27 '12 at 23:11

This is more like a comment... I am also interested in this question. My interest comes from the paper:

Kostant, B.; Sahi, S. (1991), "The Capelli Identity, tube domains, and the generalized Laplace transform", Advances in Math. 87: 71–92, doi:10.1016/0001-8708(91)90062-C

As far as I understand cones of positive matrices are very related to Jordan algebras and there is a comprehensive book (I have file of it, can send you), which in particular contains many things on Laplace transform:

Analysis on symmetric cones Jacques Faraut, Adam Korányi - 1994 - 382 pages Oxford University Press, Oxford, UK

I know this isn't an answer but I don't have enough rep to comment. I too am interested in the question. @Alexander Chervov: That book sounds interesting, I'd appreciate it if you could email me a copy (if you don't mind of course). My email address is tarmetjoe@yahoo.com. Thanks.

• Done. It is 7.5m so hope you get.... – Alexander Chervov Apr 24 '12 at 13:03
• Got it and was able to successfully open it. Thanks a lot! – Joe Tarmet Apr 28 '12 at 21:55