Research On Arbitrariness Of Two Kinds Of Symbolic Patterns

Sign pattern matrix(or Sign pattern) is one of a more active research topic incombinatorial mathematics, it has wide application in many subjects such as economics,sociology, computer science, biology and chemistry. In this paper, we study two new classesof sign pattern and prove that they are spectrally arbitrary sign pattern and minimallyspectrally arbitrary sign pattern by using Nilpotent-jacobian method and Nilpotent-centralizermethod.In chapter1, we introduce the origin, the history of development and researchsignificance on the sign pattern matrices. Some basic conceptions, relevant conclusions andmain results in this thesis are introduced.In chapter2, we describe construction method, Nilpotent-jacobian method andNilpotent-centralization method which are three methods to prove that sign pattern matricesare spectrally arbitrary.In chapter3, we present a new class of sign pattern matrix and prove that it is spectrallyarbitrary by using Nilpotent-jacobian method and Nilpotent-centralizer method, respectively.Moreover, we identify that they are minimal spectrally arbitrary.In chapter4, we present another new class of sign pattern matrix and prove that it isspectrally arbitrary by using Nilpotent-jacobian method. Finally, we identify that they areminimal spectrally arbitrary.