The Research On The Sign Pattern Of Drazin Inverse For The Sum Of Some Matrices

Qualitative properties of the sign pattern matrix are the main research content of sign pattern matrix theory, which belongs to the coinbinatorial matrix theory. P.A. Samuelson puts many econometrics models down to a Linear system, and poses a problem about qualitative analysis of the linear system. The prelude on the research of sign pattern matrix has been opened and the study of sign-solvable linear systems comes into being.Matrices of sign-solvable linear systems involve the study of signed pattern of matrix in-verse that is one of the most important research issues. In 1995, B.L. Shader introduced the conceptual matrices with signed M-P inverses, when he considered sign-solvable as the non-compatible linear system. Jiayu Shao and other scholars have made outstanding contribution-s in the study of matrices which have signed M-P inverse and have developed the study of S2NS matrix. The notion of matrices signed as Drazin inverses, first was traduced by Profes-sor Changjiang Bu, and was developed in the study of matrix with signed generalized inverses.There are some results on sign pattern of Drazin inverse for block matrices. However,the study of the sign pattern of Drazin inverse for the sum of some matrices is still a gap.Different from previous studies in this paper, we study some related topics on the sign pattern of Drazin inverse for the sum of some matrices, staring from its representations, we give those following results:(1) Give the necessary and sufficient condition for the (i,j)-entry of (P + Q)D to be zero;(2) Give some necessary conditions for (P + Q)D is signed;(3) Give the necessary condition for the (i,j)-entry of (P + Q + R)D to be zero;(4) Give some necessary conditions for (P + Q + R)D is signed.Matrices exponentiation often occurs in the representations of the Drazin inverse, and Mul-tiplicative property of sign pattern matrices relates to the sign pattern of Drazin inverse. There-fore, we give those following results:(1) Give characterizations for the structure of involutory sign pattern matrices;(2) Give characterizations for the structure of strict tripotent sign pattern matrices.