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Mathematics-Online course: Linear Algebra - Matrices - Special Matrices

Fourier Matrices

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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.
(Authors: App/Burkhardt/Höllig)
The orthogonality of the columns can easily be verified. The (complex) scalar product of the $ (j+1)$-th and the $ (k+1)$-th basis vector yields

$\displaystyle \sum_{\ell=0}^{n-1} w_n^{\ell j} \overline{w_n^{\ell k}} = \sum_\ell w_n^{(j-k)\ell} = \frac{w_n^{(j-k)n} - 1}{w_n^{j-k}-1}\, , $

and the numerator equals zero since $ w_n^n=1$.
(Authors: App/Burkhardt/Höllig)
For $ n=4$ we have $ w_4 = \exp(2\pi\mathrm{i}/4)=\mathrm{i}$, and we obtain

$\displaystyle W_4 = \left(\begin{array}{cccc} 1 & 1 & 1 & 1 \\ 1 & \mathrm... ... & -1 & 1 & -1 \\ 1 & -\mathrm{i} & 1 & \mathrm{i} \end{array}\right)\, . $

(Authors: App/Burkhardt/Höllig)
  automatically generated 4/21/2005