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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.

user127323
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    Using the hint you can reduce the case where $D$ is diagonal by a change of variables. When $D$ is diagonal, the integrand can be written as a product, and writing the integral of this product as a product of one-variable integrals, the problem reduces to the one-dimensional case. – Dan Fox Apr 24 '14 at 07:27

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