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:
The square of the Fourier transform is the flip permutation matrix:
The forth power of the Fourier transform is the identity:
Exercise (**). Proof that if N is a prime number than for any 0 < k < N
where P is a cyclic permutation matrix.
Exercise (*). If a network is rotation invariant then its Dirichlet-to-Neumann operator is diagonal in Fourier coordinates. (Hint) The harmonic functions commute w/rotation.