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Fourier Matrix

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Raising the root of unity

$\displaystyle w_n = \exp(2\pi\mathrm{i}/n) $

to higher powers we obtain the so called Fourier matrix

$\displaystyle W_n = \left(\begin{array}{ccc} w_n^{0\cdot 0} & \cdots & w_n^{0 ... ... w_n^{(n-1) \cdot 0} & \cdots & w_n^{(n-1)\cdot (n-1)} \end{array}\right)\, . $

Normalizing ( $ W_n \to W_n/\sqrt{n}$) yields a unitary matrix.

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  automatisch erstellt am 19.  8. 2013