Reducible powerful ray pattern matrices Original Research Article

A ray pattern is a matrix each of whose entries is either 0 or a ray in the complex plane originating from 0 (but not including 0). A ray pattern is a natural generalization of the concept of a sign pattern, whose entries are from the set {+, −, 0}. Powers of sign patterns and ray patterns, especially patterns whose powers are periodic, have been studied in several recent papers. A ray pattern A is said to be powerful if Ak is unambiguously defined for all positive integers k. Irreducible powerful ray patterns have been characterized recently. In this paper, reducible powerful ray patterns are investigated. In particular, for a powerful ray pattern in Frobenius normal form, it is shown that the existence of a nonzero entry in an off diagonal block implies that the corresponding irreducible components are related in a certain way. Further, the structure of each of the off diagonal blocks is characterized.