Sign Pattern Matrix

The origin of sign pattern matrix is lies in study of sign-stability and sign-solvability of linear system. It was first proposed by P.A.Samuelson in《Foundations of Economic Analysis》, and main results related it before 1995 was summarized in《Matrices of sign-solvability of linear system》by R.A.Brualdi and B.L.Shader. It gives a lot of new conclusion, So that sign pattern matrices become a new research focus in combinational mathematics. This paper is divided into four chapters, It mainly discussed idempotent, generalized inverses and spectrally arbitrary of sign pattern matrix.The author mainly introduces the history of development and research status of sign pattern matrix. Simultaneously, the author also introduce some basic definitions, relevant conclusions are introduced about idempotent, generalized inverses and spectrally arbitrary of sign pattern matrix.I discuss idempotent, generalized inverses of general sign pattern matrix.After conclude the non-negative sign pattern matrix of a number of findings,then I use similar proved methods to get some natures of the general sign idempotent pattern matrix and generalized inverses.I analyse some conclusions of spectrum arbitrary and give two sign patterns.then I prove two classes sign pattern matrix that are spectrally arbitrary using Nilpotent-Jacobi method.