I'm working on a question from Stein and Shakarchi's Fourier Analysis. This is exercise 5 from chapter 6:
Let $A$ be a $d \times d$ positive definite symmetric matrix with real coefficients. Show that $$\int_{\mathbb{R}^d} e^{- \pi (x, A(x))} dx = (\det(A))^{-\frac{1}{2}} $$
For a hint, it suggests using the spectral theorem to write $A = RDR^{-1}$, with $R$ a rotation matrix and $D$ diagonal with entires of eigenvalues of $A$.
I would really appreciate a hint to help me get started. I just don't see any way to use the hint to get the result. I'm not very familiar with integration in $\mathbb{R}^d$, so I might just be missing something obvious. Any help would be appreciated.