For an integer N, let
be the N'th root of unity, that is not equal to 1.
.
We consider the following symmetric Vandermonde matrix:
![{\displaystyle \mathbf {F_{N}} ={\frac {1}{\sqrt {N}}}{\begin{bmatrix}1&1&1&\ldots &1\\1&\omega &\omega ^{2}&\ldots &\omega ^{(N-1)}\\1&\omega ^{2}&\vdots &\ldots &\omega ^{2(N-1)}\\\vdots &\vdots &\vdots &\ddots &\vdots \\1&\omega ^{(N-1)}&\omega ^{2(N-1)}&\ldots &\omega ^{(N-1)^{2}}\\\end{bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/28e1b577ed9d93c572f6a6b86320f57e5886192a)
For example,
![{\displaystyle \mathbf {F_{5}} ={\frac {1}{\sqrt {5}}}{\begin{bmatrix}1&1&1&1&1\\1&\omega &\omega ^{2}&\omega ^{3}&\omega ^{4}\\1&\omega ^{2}&\omega ^{4}&\omega ^{6}&\omega ^{8}\\1&\omega ^{3}&\omega ^{6}&\omega ^{9}&\omega ^{12}\\1&\omega ^{4}&\omega ^{8}&\omega ^{12}&\omega ^{16}\\\end{bmatrix}}={\frac {1}{\sqrt {5}}}{\begin{bmatrix}1&1&1&1&1\\1&\omega &\omega ^{2}&\omega ^{3}&\omega ^{4}\\1&\omega ^{2}&\omega ^{4}&\omega &\omega ^{3}\\1&\omega ^{3}&\omega &\omega ^{4}&\omega ^{2}\\1&\omega ^{4}&\omega ^{3}&\omega ^{2}&\omega \\\end{bmatrix}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6393b13da8d5641b0cb59092a69dbbceb2135a8c)
The square of the Fourier transform is the flip permutation matrix:
![{\displaystyle \mathbf {F} ^{2}=\mathbf {P} .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a90cc6e8dbc4fa516fa7bc416e032f3feb35c52)
The forth power of the Fourier transform is the identity:
![{\displaystyle \mathbf {F} ^{4}=\mathbf {I} .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1e4f487804882cdcb9ae3198db74598021982e1b)
Exercise (**). Proof that if N is a prime number than for any 0 < k < N
,
where P is a cyclic permutation matrix.
If a network is rotation invariant then its Dirichlet-to-Neumann operator is diagonal in Fourier coordinates.