Two Classes New Minimal Spectrally Arbitrary Patterns

The sign pattern matrix is an important problem in the domain of the combinatorial mathematics. The research and future development of sign pattern matrix are widespread. It involves the computer science, economics, sociology, biology, chemistry, and so on. In the first two chapters, the author introduce the history of development and the related knowledge of the sign pattern matrix, and two methods that prove a sign pattern matrix is spectrally arbitrary, the structure method and Nilpotent-Jacobi method, and comparison analysis of the two methods. In the last two chapters, the author find two classes sign pattern matrices and prove they are spectrally arbitrary sign patterns using the Nilpotent-Jacobi method, and they are minimally spectrally arbitrary. In the chapter 1, the author introduce the history of development and the related knowledge of the sign pattern matrix, and the main results of the paper.In the chapter 2, the author introduce two methods that method a sign pattern matrix is spectrally arbitrary, the structure method and Nilpotent-Jacobi method with examples.In the chapter 3, the author find a sign pattern matrix with nonzero entries, and prove it is a spectrally arbitrary sign pattern using the Nilpotent-Jacobi method, and prove it is a minimally spectrally arbitrary sign pattern.In the chapter 4, the author find another sign pattern matrix with nonzero entries, and prove it is a spectrally arbitrary sign pattern using the same method, and prove it is a minimally spectrally arbitrary sign pattern.