Spectral properties of hermitean matrices whose entries are roots of unity

Abstract

Let H_n(q) denote the set of all n by n Hermitean matrices whose entries are qth roots of unity. This thesis studies the spectral properties of matrices in H_n(q) for n, q in natural number N. We determine (conjecturally sharp) upper bounds for the number of residue classes of characteristic polynomials of matrices in H_n(q), modulo ideals generated by powers of (1 - zeta), where zeta is a primitive qth root of unity. We prove a generalisation of a classical result of Harary and Schwenk on a congruence of traces modulo ideal (1 - zeta ), which is a crucial ingredient for the proofs of our main results. We also prove that, when n is odd, the switching class of each matrix in H_n(q) contains exactly one Euler graph. Lastly, we solve a problem of Et-Taoui about a potential sufficient condition for the switching equivalence of Seidel matrices.