I need to evaluate the following matrix integral:
$I = \int \mathrm{etr}\left(-cX^TX\right)\mathrm{det}(I_d + AX^TBX)^{-k/2}dX$
where $A$, $B$, and $X$ are $d\times d$, $A$ and $B$ are positive semidefinite, $c\in\mathbb{R}_{>0}$, and $k\in\mathbb{N}$.
The $d=1$ case is simple enough:
$I_{d=1} = \int_{-\infty}^\infty (1+bx^2)^{-k/2} e^{-ax^2}dx = 2\int_0^\infty (1+bx^2)^{-k/2} e^{-ax^2}dx$
Substituting $y=bx^2$ gives
$I_{d=1} = b^{-1/2}\int_0^\infty y^{-1/2}(1+y)^{-k/2} e^{-\frac{a}{b}y} dy$
which is known to be an integral representation of the confluent hypergeometric function (see 13.4.4 here). This gives the result
$I_{d=1} = \sqrt{\frac{\pi}{b}}U\left(\frac{1}{2},\frac {3-k}{2}, \frac{a}{b}\right)$
Now I want to consider the full $d\times d$ case. The confluent hypergeometric function of a symmetric $d\times d$ matrix $R$ is defined as
$\Psi(a,b,R) = \frac{1}{\Gamma_d(a)}\int_{S>0}\mathrm{etr}(-RS)\mathrm{det}(S)^{a-\frac{1}{2}(d+1)}\mathrm{det}(I_d+S)^{b-a-\frac{1}{2}(d+1)}dS$
where $\mathrm{Re}(R)>0$ and $\mathrm{Re}(a)>\frac{1}{2}(p-1)$.
Since there is a definition of the confluent hypergeometric function for matrix arguments, I would expect evaluation to be similar to the $d=1$ case. However, since I'm not very familiar with matrix calculus, I'm not sure how to proceed with the variable substitution that's analogous to $y=x^2$.
Very broadly, I want to
- Substitute $Y = AX^TBX$
- Calculate the Jacobian $J(X\to Y)$
- Invert the transformation $Y = AX^TBX$ to find an expression for $X$ in terms of $Y$ to substitute into the integral as needed
Any help with these steps would be appreciated.