Representations Of Drazin Inverse For Some Block Matrices

Generalized inverse of matrices theory is an active field in the study of matrix algebra. It has many important applications in control theory, financial mathematics and optimization and so on. There are many questions unresolved in matrix algebra. The representations of Drazin inverse, group inverse for block matrices and the existences of groupinverse are important unresolved issues.In 1979, Campbell and Meyer came up the following open problem:find an explicit expression for the Drazin inverse of a 2×2 block matrix (?)in terms of its various blocks, where the blocks A and D areassumed to be square matrices. At the present time there is no known representation for it. Some scholars only give the expressions under some special conditions. Furthermore, the representation of the Drazin (group)inverse for block matrix of the form (?)(A is square matrix) havenot been given.For A∈C^n×n, Ind(A)=k ,the matrix X∈C^n×n is said to be the Drazin inverse of A if it holds that A^kXA=A^k, XAX=X, AX=XA.Denoteby X=A^D,where k is the smallest nonnegative integer such that rankA^k+1 =rankA^k.The case when Ind(A)=1, X is called the group inverse of Aand is denoted by X=A^#.Firstly, the dissertation outlines the research significance of thegeneralized inverse of matrices and the status in domestic and overseas, Secondly, the basic knowledge is introduced. Finally, the paper studies the Drazin inverse and the group inverse of 2×2 block matrices of its special blocks in Chapter 3, 4 and 5 respectively. The following are main results:(1) Give the representations of the Drazin inverse for blockmartices(?)where E∈(?)(2) Give the existences and representations of the group inverse forblock matrices (?)and(?),where A,B∈K^n×n,A²=A.(3) Give the representations of Drazin inverse for block matrix(?),where A,B∈(?),A²=A.