# Matrix integral involving trace and determinant

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?

• Interesting, where does it come from? Mar 31, 2021 at 15:08
• Mar 31, 2021 at 15:12
• 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. Mar 31, 2021 at 15:14
• 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.) Mar 31, 2021 at 19:03
• Also, what is $\Gamma_p$? Mar 31, 2021 at 19:14