The Minimal Spectrally Arbitrary Sign Pattern Matrix

Sign pattern matrix is a very active research topic in combinatorial matrix theory, and one of the important reasons is that it has wide application in many subjects such as economics,biology,chemistry,sociology and computer science. In this paper, we characterize two classes of sign patterns which are minimally spectrally arbitrary sign patterns.In chapter 1,we introduce the history of development on the sign patters matrices,some method in our paper has used,and our research problems and main results.In chapter 2,we prove that a sign pattern matrix of order n≥5 is a minimally spectrally arbitrary sign pattern, and every superpattern of it is a spectrally arbitrary sign pattern.In chapter 3,we prove that a sign pattern matrix of order n≥6 is a minimally spectrally arbitrary sign pattern, and every superpattern of it is a spectrally arbitrary sign pattern.