Irreducible powerful ray pattern matrices

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 A(k) is unambiguously defined for all positive integers k. In this paper, irreducible powerful ray patterns, in particular, primitive powerful ray patterns, are completely characterized. Among the various characterizations, it is established that every irreducible powerful ray pattern is a ray multiple of a periodic irreducible powerful ray pattern. (C) 2002 Elsevier Science Inc. All rights reserved.