Periodic, reducible, powerful ray pattern matrices

A ray pattern (matrix) is a matrix each of whose entries is either 0 or a ray in the complex plane of the form {reiθ:r>0}{reiθ:r>0}. A ray pattern A   is said to be powerful if AkAk is unambiguously defined for all positive integers k. If A   is powerful and Al=Al+pAl=Al+p for some positive integers l and p, then we say A is periodic. In this paper, we characterize periodic, reducible, powerful ray pattern matrices and obtain new results on the powers of powerful or nonpowerful ray patterns.