Fourier transform problem with symmetric matrix. Related to Gaussian? Hi everyone I encountered a problem that looks simple enough but I have no idea where to start.

Find Fourier transform of $e^{-\langle Ax,x\rangle}$ when $A$ is a positive definite symmetric $n \times n$ matrix. 

Since $A$ can be any symmetric positive definite matrix, I have no idea how to apply Fourier transform formula I learned in class. I does look like Gaussian, but I really need some help. 
Help would be very much appreciated. Thank you!
 A: $U^tAU=D$ where $D$ is diagonal and $U^tU=I$. Use $U$ to change variables in the multi-dimensional transform integral. Starting with
$$
            \int_{\mathbb{R}^n}e^{-(Ax,x)}e^{ix\cdot\xi}dx,
$$
Let $x=Uy$. Because $U$ is orthogonal, then $\|x\|=\|Uy\|=\|y\|$ gives a constant Jacobian equal to $1$. Therefore, the above is transformed to
$$
           \int_{\mathbb{R}^n}e^{-(U^tAUy,y)}e^{iy\cdot(U^t\xi)}dy
   = \int_{\mathbb{R}^n}e^{-(Dy,y)}e^{iy\cdot U^t\xi}dy \\
   = \left.\int_{\mathbb{R}^n}e^{-\alpha_1 y_1^2}e^{-\alpha_2 y_2^2}\cdots e^{-\alpha_n^2y_n^2}e^{iy\cdot\eta}dy\right|_{\eta=U^t y}.
$$
The $\alpha_j$ are the diagonal elements of the diagonal matrix $D$, which are the eigenvalues of $A$. By assumption $\alpha_j > 0$ for all $j$. So everything reduces to single variable Gaussian transforms, with a final evaluation at $\eta=U^t \xi$.
A: Hint: Since $A$ is an SPD matrix, it has a unique SPD square root.  Apply the change of variables
$$
y = A^{1/2}x
$$
Noting that
$$
\langle Ax,x \rangle = 
\langle A^{1/2}x,A^{1/2}x \rangle
$$
