TITLE: Fourier transform of a multivariate gaussian
QUESTION [5 upvotes]: What is the Fourier transform of the following function:
$$
    f(\mathbf{x}) = \mathbf{e}^{-\frac{1}{2}\mathbf{x}^T \mathbf{P} \mathbf{x}}
$$
I know the solution for the one-dimensional problem, and I can find the solution for $\mathbf{P} = \mathbf{I}$, but I don't know how to handle a general positive-definite $\mathbf{P}$.

REPLY [2 votes]: Since $\mathbf P$ symmetric  positive definite, $\mathbf P=\mathbf Q^T\,\mathbf D\,\mathbf Q$ where $\mathbf D$ is diagonal and $\mathbf Q$ is orthogonal. Then
$$
\mathbf{x}^T\, \mathbf{P}\, \mathbf{x}=(\mathbf{Q}\,\mathbf{x})^T\,D\,(\mathbf{Q}\,\mathbf{x}).
$$
Make the change of variables $\mathbf y=\mathbf{Q}\,\mathbf{x}$, and take into account that since $\mathbf Q$ is orthogonal
$$
\mathbf x\cdot\mathbf\xi=\mathbf y\cdot(\mathbf Q\,\mathbf\xi).
$$