Eigenvalues and eigenvectors of the finite Fourier transform

The eigenvalues and eigenvectors of the n×n unitary matrix of finite Fourier transform whose j, k element is (1/(n)1/2)exp[(2πi/n)jk], i=(−1)1/2, is determined. In doing so, a multitude of identities, some of which may be new, are encountered. A conjecture is advanced.