Spectra of Hadamard matrices

In this paper, we develop a technique for controlling the spectra of Hadamard matrices with sufficiently rich automorphism groups. For each integer t >= 2, we construct a Hadamard matrix H-t equivalent to the Sylvester matrix of order n(t) = 2(2t-1-1) such that the minimal polynomial of 1/root nt H-t is the cyclotomic polynomial Phi(2t+1)(x). As an application we construct real Hadamard matrices from Butson Hadamard matrices. More concretely, a Butson Hadamard matrix H has entries in the kth roots of unity and satisfies the matrix equation HH* = nI(n). We write BH(n, k) for the set of such matrices. A complete morphism of Butson matrices is a map BH(n, k) -> BH(m, l). The matrices H-t yield new examples of complete morphisms BH(n, 2(t)) -> BH(2(2t-1-1)n, 2), for each t >= 2, generalising a well-known result of Turyn.