If $A$ is a $n\times n$ symmetric positive definite matrix, I would like to compute this integral:
$\int_{\mathbb R^n} \|x\|^2 e^{-x\cdot Ax} dx$.
I have tried to use a Cholesky decomposition for $A = L^TL$. Then, applying the change of variables formula I get that it is the same as:
$\frac{1}{\sqrt{det(A)}} \int_{\mathbb R^n} \|Lu\|^2 e^{-\|u\|^2}du$.
It seems to be a simpler form, but I don't know what to do from here. Any help would be appreciated. Thank you!