Constructing symmetric nonnegative matrices via the fast fourier transform

We derive an algorithm based on the fast Fourier transform to construct a real symmetric matrix S with eigenvalues λ1≥λ2≥ ≥λn, with eigenvector e = [1, 1, … ,1]T belonging to the eigenvalue λ1.We find simple conditions on the eigenvalues such that the algorithm constructs an irreducible matrix S = λ1E, where E is a symmetric doubly stochastic matrix.