To simplify the question (and reduce to a more classical one), you could consider diagonalizing .

This is not always possible, but for every , , hence you can replace by , which is real and symmetric. As a consequence, diagonalizes in an orthonormal basis. Let be its eigenvalues. Changing from one orthonormal basis to another doesn't change the volume (this is a rotation, perhaps plus a symmetry, anyway it is an orthogonal transformation), hence your integral becomes, using this change of basis:

.

(I swapped the factors to avoid confusion) You can expand the first sum so that you get the sum of integrals, and then integrate each of them rather simply. I think the integral converges iff the 's are positive.

One question remains: what is the connection between the 's and ? Don't you in fact assume that itself is symmetric (and positive)?

I think what you finally get is: , which is also . The previous question becomes: for which does the integral converge? (or: for which is positive?) I don't know the answer. Does anyone?