Special Spectrally Arbitrary Pattern

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 four sign patterns which are minimally spectrally arbitrary sign patterns and a class of spectrally arbitrary sign patterns.In chapter 1,we introduce the history of development on the sign pattern matrices, some method in our paper had used,and our research problems and main results.In chapter 2,we discuss that sign patterns A and B are minimally spectrally arbitrary sign patterns,and every superpattern of them is a spectrally arbitrary sign pattern.In chapter 3,we discuss that another two sign patterns C and D are minimally spectrally arbitrary sign patterns,and every superpattern of them is a spectrally arbitrary sign pattern.In chapter 4,we characterize a class of new spectrally arbitrary sign patterns Fn,r,where the positive entry in the last row of Fn,r is in rth column,2≤r≤n.and every superpattern is a spectrally arbitrary sign pattern.