Spectrum Of Arbitrary Pattern

Sign pattern matrix is a very active research topic in combinatorial mathematics,oneof the important reasons is that the study of sign pattern matrix in economics, biology,chemical, sociological and theories of computer science have extensive practical application.In this paper, we characterize two classes of sign patterns which are spectrally arbitrary signpatterns.In chapter 1, we introduce the history of development on the sign pattern matrix, therelated knowledge of the spectrally arbitrary sign pattern matrix, and the main results ofthe paper.In chapter 2, we discuss a class of n by n (n≥6)sign patten matrices A with2n + 1 nonzero entries,and find the necessary and su?cient conditions of A to be spectrallyarbitrary sign pattern.In chapter 3, we characterize a class of n by n(n≥5)sign patten matrices with 2nnonzero entries are spectrally arbitrary by using the Nilpotent-Jacobian method.We prove itis actually minimal spectrally arbitrary pattern and every super-pattern of it is a spectrallyarbitrary sign pattern.