- #1

Aquinox

- 10

- 0

## Homework Statement

Let A be a real, symmetric positively definite nxn - matrix.

[tex]f:\mathbb{R}^{n}\rightarrow\mathbb{R}\; s.t\;\vec{x}\rightarrow e^{-\frac{1}{2}<\vec{x},A\vec{x}>}[/tex]

Show that the FT of f is given by:

[tex]\hat{f}(\vec{k})=\frac{1}{\sqrt{\det A}}e^{-\frac{1}{2}<\vec{k},A^{-1}\vec{k}>}[/tex]

## Homework Equations

If I'm not very much mistaken:

[tex]\hat{f}(\vec{k})=\int_{\mathbb{R}^{n}}f(\vec{x})e^{-2\pi i<\vec{k},\vec{x}>}d^{n}x[/tex]

## The Attempt at a Solution

Quite honestly I have no idea anymore. I suppose I'm missing sth. quite trivial.

I've tried to change <x,Ax> to [tex]x^tAx[/tex] and doing the same with <k,x> and then multiplying from right by A^-1*x and some more but kept running in circles.

I'm terrible with this matrix-stuff and on the solution of this task depends the solution of another one