Powerful ray patterns

Jong Sam Jeon

doi:10.7273/000005734

Since the concept of ray pattern was introduced, many authors have studied properties of ray pattern. At the _rst appearance of ray pattern, authors considered numerical properties of complex matrices by using the concept of ray pattern. In this sense, a ray pattern can be considered as a abstraction of complex matrices. On the other hand, there had been numerous studies on combinatorial properties of sign patterns. Hence extension from sign patterns to ray patterns was very natural to get more generalized results in combinatorial matrix theory. So a ray pattern has two aspects; an abstraction of a complex matrix and a generalization of a sign pattern. In this thesis, we are going to think about a certain combinatorial property of ray patterns. Ray patterns which we are most interested in in this thesis behave well under powers, called powerful ray patterns, in the sense that any power of a given ray pattern does not have ambiguous entries. Also we are going to consider the set S. A ray pattern is in S if it is ray diagonally similar to a ray multiple of Boolean pattern of itself. We are going to address three questions and answer them partially or fully in this thesis. Those questions are characterizing powerful ray patterns, checking powerfulness of irreducible ray patterns by powering, and characterizing the set S. The _rst question is still open in general case. We are going to answer this question for ray patterns whose diagonal blocks of Frobenius normal form are primitive. For the second question, we are going to see an answer which gives us an upper bound on the _rst power that a non-powerful ray pattern will encounter an ambiguous entry. This answer does not cover every possible cases but exceptional cases are very specialized. For the last question, we are going to see two complete answers by using products of chains and powers of a certain matrix. Furthermore, we are going to have an algorithm that checks if a given ray pattern is in S or not by combining those two answers. At the end of this thesis, we are going to see examples of ray patterns which are not considered in this thesis. Those examples illustrate three possible cases of ray patterns that are reducible and non-powerful. We hope that studying those three cases would lead us to a complete answer for characterizing powerful ray patterns.

Powerful ray patterns

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