Some Limit Results Of Group (Drazin) Inverses For The Block Matrices

Generalized inverses of matrices have very important applications in optimization theory, population growth model, Numerical linear algebra, Markov chains, differential equation, etc. So, the research on the representations of generalized inverses of the matrix is very important. And the representations of the group and Drazin inverse of the block matrices are researched more active by the scholars.Let Cm×n be the set of all m×n matrices over the complex number field. For A∈Cn×n, Ind (A)=k. The matrix X∈Cn×n satisfying the following equations Ak XA=Ak, XAX=X, AX=XA is called the Drazin inverse of A, denoted by X=AD or X=Ad. Particularly, when Ind (A)≤1, AD is called the group inverse of A, denoted by A#In1974, CD. Meyer gave the representation of the Drazin inverse of the upper triangular block matrix with the limit, i.e.(?)(Rp+1+εI)-1Rp=RD, where R∈Cn×n, non-negative integer p≥Ind(R). Then, in1977, Meyer gave the limit formula of the group inverse of M, where M∈Cn×n,M#exist (If M#exist, then M#=(?)(M2+tI)-1M)(see [4]). In this paper, we give some limit results of group (Drazin) inverses, and also give a method to compute group inverses for the block matrices.In this paper, Chapterl introduces the research back ground, the research status, the status in domestic and overseas of generalized inverses of matrices; Chapter2gives the basic knowledge related to this paper; Chapter3and Chapter4give the research results of this paper respectively, the main results list as follows.1. IfM∈Cn×n, k=Ind(M), then MDM=(?)(Mk+tI)-1Mk=(?)(M+tI)-kMk2.IfA,B∈Cn×n, A#exists, then (?)(tI-B(A+tI)1A)1B=-B(AA#B)#Moreover, using these results, we give a method to compute group inverses for the block matrices, and also give some examples as applications.