Inertia And Rank Of Sign Pattern Matrices

Sign pattern matrix is a very hot research topic in combinatorial matrix theory, for which it is of greate value for practical applications in many academic subjects such as economics, biology, chemistry, computer science, etc. It was first proposed by P.A.Samuelson in《Foundations of Economic Analysis》. He pointed out that the solutions to some questions in economic and other fields rely on matrix element sign rather than its dimension. Research on sign pattern matrix, in a sense, is equal to qualitative analysis of matrix. R.A.Brualdi, B.L. Shader summarized all the vital achievements by 1995 in his monograph《Matrices of Sign-solvability of Linear System, thus turned a new page of the study. Since 1995, there appears a mass of articles and notions, such as ray sign pattern, etc.The paper mainly discusses some special forms of sign pattern matries.In chapter one, 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 two, it includes some impontant forms of sign pattern matrix inertias and research trends of inertia.In chapter three, it includes prelimary investigation of mininum rank of sign pattern matrix and some study taken by the author.