- #1
Aquinox
- 10
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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