I am trying to evaluate the integral

$$\int\_{\mathbb{R}^d} e^{i \langle z,x \rangle} e^{- \langle x,A^{-1}x \rangle /2} dx $$

where $A$ is a positive-definite symmetric matrix. 

As a first step (which is quite pointless, but just to remove the inverse sign) I did

$$ |\det A|\int\_{\mathbb{R}^d} e^{i \langle z, Ax \rangle} e^{- \langle A x,x \rangle /2} dx . $$

What I really want to do now is make some orthogonal change of variables so that all the matrices inside the integral are diagonal. However, I cannot make this calculation work out, regardless of what I try.

Note that it is possible to diagonalize $A$ using an orthogonal matrix, given the hypotheses on $A$.

Many thanks for your help.

EDIT:

Let

$$f\_X(x) = \frac{1}{\sqrt{(2\pi)^d\operatorname{det}(A)}}\exp\left(-\frac{1}{2}\langle x, A^{-1}x\rangle\right).$$

We calculate as follows (as per the suggested substitution):

\begin{align}
 \int\_{\mathbb{R}^d} e^{i \langle z,x \rangle} f\_X(x)dx &= \frac{1}{\sqrt{(2\pi)^d\operatorname{det}(A)}} \int\_{\mathbb{R}^d} e^{i \langle U^T z,y \rangle} e^{\langle y, D^{-1}y \rangle /2} dy \\
&= \frac{1}{\sqrt{(2\pi)^d\operatorname{det}(A)}} \prod\_{n=1}^{d} \int\_{\mathbb{R}} e^{i (U^T z)\_n y\_n} e^{-y\_n^2/2d\_n} dy\_n
\end{align}

where, for a vector $x \in \mathbb{R}^d$ I write $x = (x\_1,...,x\_d)$, and where the $d\_n$ for $n \in \{1,...,d\}$ are the diagonal entries of $D$.

Continuing from above, using the integral from the $1$-D case, we get

\begin{align}
&= \frac{1}{\sqrt{(2\pi)^d\operatorname{det}(A)}} \prod\_{n=1}^{d} \exp \left( -\frac{1}{2} d\_n (U^T z)^2\_n \right) (2 \pi d\_n)^{1/2}\\
&=\prod\_{n=1}^{d} \exp \left( -\frac{1}{2} d\_n (U^T z)^2\_n \right)\\
&= \exp \left( -\frac{1}{2} \langle z, Az \rangle \right),
\end{align}

as expected.