Representations Of Drazin Inverses For Block Matrices With Zero Schur Complement And Applications

Drazin inverses of block matrices have important applications in numerical analysis, Markov chains,differential equation,difference equation,cryptography and control theory, etc..So the research on the representations of Drazin inverses have very important significance.In 1979,Campbell and Meyer discuss applications of AD in the solution of constant coefficient differential equation,proposed an open problem to find an explicit representation of the Drazin inverse of block matrix(?) (A and D are square).Because of the difficulty of this problem,until now,the representation has not been given..In the past decades,scholars studied the representations of the Drazin inverse of(?) under some special conditions.In this dissertation,we give the representations of the Drazin inverse of the block matrix(?),under condition S=D-CAdB=0.In Chapter 1 we introduce the research significance of generalized inverses of matrices, the status in domestic and overseas,and the basic knowledge in the theory of generalized inverses of matrices;in Chapter 2 we introduce the concepts,properties,construct methods of M-P inverse,{i,j,k} inverse,and the applications of all kinds of generalized inverses;in Chapter 3 we introduce the main results of this paper,which are listed below:1.The representation of the Drazin inverse of the block matrix(?) are given,under conditions S=D-CADB=0,ADBC=0,AπABC=0.2.The representation of the Drazin inverse of the block matrix(?) are given, under conditions S=D-CADB=0,ABCA=0,CAAπ=0,CAπB=0.3.The representation of the Drazin inverse of the block matrix(?) are given, under conditions S=D-CADB=0,ADBC=0,CAAπ=0.4.The representation of the Drazin inverse of the block matrix(?) are given, under conditions S=D-CADB=0,ADBC=0,AAπB=0. 5.The resentation of the Drazin inverse of the block matrix(?) are given, under conditions S=D-CADB=0,AπBCAD=0,AAπBC=0,CAπBC=0.6.The representation of the Drazin inverse of the block matrix(?) are given, under conditions S=D-CADB=0,ADBCAπ=0,BCAπA=0,BCAπB=0.7.The rpresentation of the Drazin inverse of the block matrix(?) are given, under conditions S=D-CADB=0,ABC=0,CAπ=0.