The eigenstructure and Jordan form of the Fourier transform over fields of characteristic 2 and a generalized Vandermonde-type formula

In this paper, we describe the eigenstructure and the Jordan form of the Fourier transform matrix generated by a primitive N-th root of unity in a field of characteristic 2. We find that the only eigenvalue is λ=1 and its eigenspace has dimension [N4]+1; we provide a basis of eigenvectors and a Jordan basis. The problem has already been solved, for number theoretic transforms, in any other finite characteristic. However, in characteristic 2 classical results about geometric multiplicity do not apply and we have to resort to different techniques in order to determine a basis of eigenvectors and a Jordan basis. We make use of a modified version of the Vandermonde's formula, which applies to matrices whose entries are powers of elements of the form x+x−1.