In a paper by Mathai, he uses the following integral representation of a determinant, (or, really, what I give is a simple special case of what he gives), without any explication. All matrices are real $p\times p$ symmetric positive definite.

\begin{equation} | I-U |^{-a} = \frac{1}{\Gamma_p(a)} \int_{T>0} |T|^{a-(p+1)/2} \exp(-\text{Tr}(I-U)T) \;dT \end{equation}

where $U$ satisfies $0\lt U \lt I$ (in the cone order in the cone of positive definte matrices), the integral is over the cone of positive definite matrices and $\Gamma_p(a)$ is the generalized gamma function in dimension $p$, and $\Re(a) > (p-1)/2$.

Any references for this?

  • $\begingroup$ Alexander: I do not understand your simpler example, I get $\frac{1}{\text{Tr}M}$. The role of the integer $p$ is as the dimension of the positive-definite matrix$U$ and $T$. $\endgroup$ – kjetil b halvorsen Jun 21 '12 at 21:45
  • $\begingroup$ @Kjetil B Halvorsen Sorry I was in hurry... $\endgroup$ – Alexander Chervov Jun 24 '12 at 7:26

OLDER EDIT. (Elementary derivation) I realized that my original answer was actually overkill for the question. The said integral in question follows from the definition of the multivariate Gamma function

\begin{equation*} \Gamma_p(a) := \int_{A > 0} \exp(-\mbox{tr}(A))\det(A)^{a-(p+1)/2}(dA), \end{equation*} where $\Re(a)>(p-1)/2$.

From this it follows (by a change of variables) that for a positive definite matrix $S$, \begin{equation*} \int_{A > 0} \exp(-\mbox{tr}(S^{-1}A))\det(A)^{a-(p+1)/2}(dA) = \Gamma_p(a)\det(S)^a, \end{equation*} so that with $S=(I-U)^{-1}$ we obtain the integral in question.

Of course, to complete the picture it may be helpful to express $\Gamma_p(a)$ in more elementary terms. Chapter 2 of Muirhead's book provides these details. I cite the result that provides this expression.

Theorem (Muirhead (1982), Thm 2.1.2) Let $\Re(a) > (p-1)/2$. Then, $$ \Gamma_p(a) = \pi^{p(p-1)/4}\prod_{j=1}^p \Gamma(a - (j-1)/2) $$

(Hint: To prove the above, write the Cholesky decomposition $A=T'T$, with these change of variables, the original Gamma integral factorizes into a product of Gaussian and Gamma integrals.)

The part that I recalled below provides yet another representation that expresses the multiplicative determinantal lhs in terms of an infinite sum.


This is actually somewhat classical knowledge. Here are two related pointers.

A partition $\tau=(t_1,\ldots,t_m)$ is a vector of nonnegative integers listed in increasing order, and $|\tau|$ denotes $t_1+\cdots+t_m$. The generalized Pochhammer symbol $(a)_\tau$ is defined as \begin{equation*} \newcommand{\risingf}[2]{{{#1}}^{\overline{{#2}}}} (a)_\tau := \frac{\Gamma_d(a+\tau)}{\Gamma_d(a)} = \prod_{l=1}^m \risingf{\bigl(a - \tfrac{1}{2}(l-1)\bigr)}{t_l} \end{equation*}

Let $C_\tau(X)$ be the Zonal Polynomial with signature partition $\tau$. Then, the following representation exists

For a matrix $U$ satisfying $\|U\| < 1$, we have the following "binomial-theorem"

\begin{equation} \frac{1}{|I-U|^a} = \sum_{k\ge 0}\sum_{|\tau| = k} \frac{(a)_\tau C_\tau(U)}{k!}. \end{equation}

Using representations for these Zonal polynomials, one can obtain the integral representation mentioned in the original post.

More directly, you can look at Chapter 7 of R. Muirhead, "Aspects of Multivariate Statistical Theory", where you'll see that actually, $|I-U|^{-a}={}_1F_0(a;U)$, a matrix argument hypergeometric function. I've to run now, if I get a chance I'll clean up my answer and fill in the details.

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A very similar (but different) expression for log of the determinant is given by Du and Ji in their paper "an integral representation of the determinant of a matrix and its applications". I am guessing a slight adaptation of their thing can get your formula.

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