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**:

For example,

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.