The Sign Pattern Of Drazin Inverse Of Matrix

Generalized inverse is widely used in both science and engineering field.The study of sign pattern of generalized inverse is important for the theoretical and application significance in qualitative analysis and combinatorial matrix theory.P.A.Samuelson,Nobel Prize laureate in economics,concluded the economic models as the problem of sign solvable of linear system,and opened the research field of sign pattern.After that,R.A Brualdi,AMS fellow,and B.L Shader,SIAM fellow,has systematic studied the sign pattern matrix and developed the early theory for the research of sign pattern of generalized inverse.From 1995-2003,B.L shader and Shao et al,have studied Moore-Penrose and demonstrated some important results such as the matrix with signed M-P inverse.In 2011,Prof.Bu proposed the concept of sign group invertible matrix,strong sign invert-ible matrix and the matrix with signed Drazin inverse.Study of the Drazin inverse of matrix is a cutting-edge research in this field and will open a pathway to understanding the sign pat-tern matrix.There are still many questions need to be solved by developing new theory and method.At frist,this paper introduces the theory of sign matrix,the recent research progress of sign pattern matrix,the research status and the basics knowledge of generalized inverse and sign generalized inverse.Based on these theories,we give some results on upper triangular block matrix(A/O B/C)and anti triangular block matrix(A/C B/O)with signed Drazin inverse.In the third chapter,according to the representation of the Drazin inverse of block trian-gular matrices,we study the structure of the block triangular matrices(reducible matrix)with signed Drazin inverse,and we give some necessary conditions of block triangular matrices having signed Drazin inverse.Besides,we also obtain some combinatorial properties of block triangular matrices with signed Drazin inverse.Based on these results,we give the structure characterizations of block triangular matrices symbols with signed inverse.In the final chapter,we focus on the study of the sign pattern of group inverse of anti triangular block matrix.In order to study the sign pattern of the group inverse of anti triangular block matrices,we first give a new the representation of group inverses of anti triangular block matrix.Based on this representation,we give the sufficient and necessary conditions of(A/C B/O)be the S2GI matrix under some conditions.