Periodic, irreducible, powerful ray pattern matrices

A ray pattern is a complex matrix each of whose entries is either 0 or a ray eiθ, where θ is a real number. For a ray pattern A = [ast], we define the ray pattern ∣A∣=[ast′] of A, where ast′=1 if ast ≠ 0 and ast′=0 if ast = 0. In this paper, we first show that an irreducible powerful ray pattern A is ray diagonally similar to ω∣A∣ for some ray ω. By using this representation, we obtain several results on irreducible powerful ray patterns and irreducible periodic ray patterns. Then we show that the number of such rays ω is k(A), where k(A) is the index of imprimitivity of A. As an application to complex matrices, we generalize the Perron-Frobenius Theorem to a subclass of complex matrices.