Reducible powerful ray pattern matrices

• Published in: Linear algebra and its applications Vol. 399; pp. 125 - 140
• Main Authors: Li, Zhongshan, Hall, Frank J., Stuart, Jeffrey L.
• Format: Journal Article Conference Proceeding
• Language: English
• Published: New York, NY Elsevier Inc 01.04.2005; Elsevier Science
• Subjects: Algebra; Exact sciences and technology; Linear and multilinear algebra, matrix theory; Mathematics; Powerful ray pattern; Ray pattern; Sciences and techniques of general use; Sign pattern; Unambiguously defined powers; 15A57; Sign pattern; Unambiguously defined powers; Ray pattern; Powerful ray pattern; 15A21; 15A48; Reducibility; Normal form; 15A57 Ray pattern; Irreducible operator; Diagonal matrix; Sign; Block matrix; Ray theory
• ISSN: 0024-3795; 1873-1856
• DOI: 10.1016/j.laa.2004.08.029

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. 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.