Let $\mathbf{A}$ be symmetric positive definite matrices of dimension $(p \times p)$ and $a$, $b$ be positive reals.

I know that the following equality should hold $$ \int_{\mathbf{A} > 0} \mathbf{A} \, \text{det}(\mathbf{A})^\frac{a - p - 1}{2}(b + \text{trace}(\mathbf{A}))^{-\frac{b+ap}{2}} d\mathbf{A} = \frac{\Gamma_p(a/2) \Gamma(b/2) b^{b/2+1} a}{\Gamma((b+ap)/2) (b-2)} \mathbf{I}. $$

Has anyone seen this integral before and knows where I can find out more about it and related integrals?

  • 3
    $\begingroup$ Interesting, where does it come from? $\endgroup$ Mar 31, 2021 at 15:08
  • 2
    $\begingroup$ Take a look at mathoverflow.net/questions/287794/… and mathoverflow.net/questions/287866/… $\endgroup$ Mar 31, 2021 at 15:12
  • $\begingroup$ A is a random matrix. Left hand side is A times kernel of pdf(A), right hand side comes from normalizing constant and expected value of the distribution. $\endgroup$
    – stollenm
    Mar 31, 2021 at 15:14
  • $\begingroup$ If we consider scalar $A$ instead, then this would seem to be a variation on the usual integral representation for the beta function. As such it seems reasonable to think of this as a sort of matrix beta integral. Those do appear to have been considered historically, e.g., the preprint of Neterin at arxiv.org/abs/1411.2110. (I'll glance and see if that may cover your case.) $\endgroup$ Mar 31, 2021 at 19:03
  • $\begingroup$ Also, what is $\Gamma_p$? $\endgroup$ Mar 31, 2021 at 19:14


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