Two Types Of The Minimal Spectrally Arbitrary Sign Pattern

Sign pattern matrix is not only a very important foundational problem in combinatorial matrix theory, but also a active subject in the science field nowadays. The research and future development of the sign pattern matrix are widely used. And it has a wide application in many subjects such as economics, biology,chemistry,sociology and computer science. In this paper we firstly introduced the history and background of development on the sign pattern matrix. We also showed some basic knowledge and main results of the sign pattern.In chapter 2, we introduced two main methods:the structure method and the Nilpotent-Jacob method.Both of them can prove whether a sign pattern matrix is spectrally arbitrary. Then we compared the two methods through several spectrally arbitrary sign patterns from references.In chapter 3 we discovered one kind of special minimally spectrally arbitrary sign pat-terns by using the method of N-J. And in chapter 4, we also found out another kind of minimally spectrally arbitrary sign patterns which need to discuss under some certain con-ditions by using the same method in chapter 3.