The Spectrum Of Arbitrary Sign Pattern Matrices

The sign pattern matrix is a very active research direction in the domain of combi-natorial mathematics in recent years,it has important application in many subjects suchas economics!biology!chemistry!sociology!computer science and so on.In this paper,wemainly use the Nilpotent-Jacobian method to prove that a sign pattern matrix and a complexsign pattern matrix are spectrally arbitrary.In chapter 1,we introduce the history of development on the sign pattern matrices, somemethods used in our paper and our research problems and main results.In chapter 2,we prove a sign pattern matrix of order n≥8 is a minimally spectrallyarbitrary sign pattern,and every superpattern of it is a spectrally arbitrary sign pattern.In chapter 3,we extend the Nilpotent-Jacobian method to prove a complex sign patternmatrix is a spectrally arbitrary sign pattern,and every superpattern of it is a spectrallyarbitrary sign pattern.