Special Spectrum Any Mode And Inertia Arbitrary Mode

The sign pattern matrix is not only a foundational problem in the domain of combinatorial mathematics, but also an important problem. The research and future development of sign pattern are widespread. It is useful to solute the problems about combination theory of matrices, combination theory of numbers, biology, chemistry, economy and so on. In this paper, we firstly compared the structure method with the Nilpotent-Jacobi method. Both of them can prove whether a sign pattern matrix is spectrally arbitrary. Then we discovered two kinds of special minimally spectrally arbitrary sign patterns using the method of Nilpotent-Jacobi. Finally, we portrayed one kind of inertially arbitrary pattern but not spectrally arbitrary pattern.In chapter 2, we compared the structure method with the Nilpotent-Jacobi method through several spectrally arbitrary sign patterns from references. The use of structure method needs great skill, so this method is only suitable for the minority of sign patterns. The most commonly used method of proving whether a sign pattern is spectrally arbitrary is the Nilpotent-Jacobi method. The key of this method is to find out a nilpotent matrix that belongs to the qualitative matrix class of that given sign pattern and to calculate whether the Jacobi on the nilpotent points is zero. When looking for some nilpotent matrix, we sometimes easily find out the nilpotent matrix. But sometimes, it is not easy to extract the nilpotent matrix, at this time, we only need proving there is some nilpotent matrix in the qualitative matrix class.In chapter 3, we find out a minimally spectrally arbitrary sign pattern mainly using the Nilpotent-Jacobi method. While in the process of looking for the nilpotent matrix, because it is not easy to extract the accurate nilpotent matrix, we can only consider the signs of these unknown quantities by using the Intermediate Value Theorem. Then we prove the Jacobi is not zero. According to the Nilpotent-Jacobi method, we know the sign pattern is spectrally arbitrary. Furthermore, we proved it is a minimally spectrally arbitrary according to the definition of minimally spectrally arbitrary. In chapter 4, we similarly obtain another minimally spectrally arbitrary sign pattern by using the Nilpotent-Jacobi method and the Intermediate Value Theorem when finding the nilpotent matrix from the qualitative matrix class of this sign pattern.If a sign pattern matrix is spectrally arbitrary, then it is inertially arbitrary. Otherwise, it is not true. In chapter 5, we investivated an inertially arbitrary sign pattern but not potentially nilpotent (certainly not spectrally arbitrary) using the methods of induction and the recursion.