Special Sign Pattern Matrix

The matrix theory of sign patterns mainly studies its qualitative nature of the sign pattern which is only about its elements. The sign pattern matrix originated some issues in the economics of the qualitative nature of the study. The research on sign pattern matrices is from some questions in the sign-solvability of a linear system and sign stability. Pioneering work in the sign pattern matrices presented by the Nobel Economics Prize winner P. Samuelson who pointed to the needed to solve certain problems in economics. In 1995, the book about Matrices of Sign—solvable Linear systems worked by R.A.Brualdi and B.L.Shade, comprehensively and systematically summed up the research results on sign patterns, meanwhile, obtained a number of new conclusions. So that sign pattern matrices become a new research focus.A matrix ( )A = ai j n×n, whose entries ai j are from the set {+ , ?,0} is called a sign pattern matrix (sign pattern). If A = A2 holds, then A is called sign idempotent sign pattern matrix. If there exists a real matrix ( )B = bi j n×n, who's sign of entries bi j are same with ai j, so that B 2 =B, then A is called A allows real idempotent. Further, if A = AT, then A is called a symmetric sign pattern. In this paper, we mainly study the sign idempotent of sign patterns, symmetric sign patterns and their relations, especially the number of negative entries that occur in sign idempotent sign pattern. In section 1, we introduce the study of sign pattern matrix of the historical background, give sign pattern matrix of the basic concepts and relevant conclusions, outline the sign pattern matrix of research and major research issues. In section 2, we study the relations between the sign idempotent and the allowance of idempotent, and then mainly characterize the idempotent of principal submatrices and find a class of sign idempotent sign patterns whose upper (lower) triangular matrices are still sign idempotent. Meanwhile, we give two structures of the sign idempotent by the definition of idempotent. In section 3, we determine the maximum number of negative entries that occur in sign idempotent sign patterns as well as symmetric sign patterns, and provide the sign pattern that achieves this maximum number. Further, we investigate generalized permutationally similar idempotent configuration and minimum rank factorization in symmetric sign pattern. we consider the generalized permutationally similar idempotent structure and minimum rank factorization. Finally, we obtain some equivalent propositions about generalized inverses, the allowance of tripotent and the allowance of idempotent.