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A matrix F is given: $$ F = [e^{i\frac{2\pi kl}{n}}]_{k,l=0}^{n-1} $$ Find $$ F^{-1} $$

I know Gaussian method for inverting matrices but I suppose it doesn't apply to matrices with not given exact n value. Could you tell me what are the methods for inverting matrices like this?

wisniak
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  • Maybe, the inverse can be calculated depending on $n$. Then, the limit, if it exists, would be the inverse of the infinite matrix. – Peter Oct 21 '14 at 20:49
  • The matrix seems to be $n\times n$, not infinite. –  Oct 21 '14 at 20:51
  • Of course, edited – wisniak Oct 21 '14 at 20:55
  • Well, this matrix is the one you use to perform a discrete Fourier transform. The inverse is then, not surprisingly, the one used when performing the inverse Fourier transform. Try to multiply $F$ by its conjugate transpose. You should get a scalar matrix. – Jyrki Lahtonen Oct 21 '14 at 21:01
  • I'd try to write this matrix for small $n$, invert in manually by any known method, and then make a hypothesis on the general form. – TZakrevskiy Oct 21 '14 at 21:01
  • For heaven's sake don't try to find the inverse by hand. – Jyrki Lahtonen Oct 21 '14 at 21:02

2 Answers2

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HINT:

This is a famous matrix, $1/\sqrt{n}\cdot F$ is unitary. One can check this directly or look at

http://en.wikipedia.org/wiki/Discrete_Fourier_transform#Orthogonality

orangeskid
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Hint: It's a Vandermonde matrix (link1 and link2).

ir7
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