User:DVD206/Fourier coordinates

For an integer N, let $$\omega$$ be the N'th root of unity, that is not equal to 1.
 * $$\omega^N = 1, \omega \ne 1$$.

We consider the following symmetric Vandermonde matrix:


 * $$\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} $$

For example,


 * $$\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}. $$

The square of the Fourier transform is the flip permutation matrix:
 * $$\mathbf{F}^2 = \mathbf{P}.$$

The forth power of the Fourier transform is the identity:
 * $$\mathbf{F}^4 = \mathbf{I}.$$

Exercise (**). Proof that if N is a prime number than for any 0 < k < N


 * $$\mathbf{F(\omega^k)} = \mathbf{P}\mathbf{F(\omega)}\mathbf{P}^T$$,

where P is a cyclic permutation matrix.

If a network is rotation invariant then its Dirichlet-to-Neumann operator is diagonal in Fourier coordinates.