Existence Conditions And Representations Of Group Inverse For Some Block Matrices

Let Kn×n denote the set of the n×n matrices over skew field K. For A∈Kn×n, the matrix X∈Kn×n is called the group inverse of A, if it holds that AXA=A, XAX=X, AX=XA.We denote X by A#. It is well-known that A#exists, it is unique. If A#exists, we suppose Aπ=In-AA#, where In is the n×n identity matrix.The group (Drazin) inverses of block matrices have important applications in singular differential equations and singular difference equations, Probability and Statistics, Markov chains, Game theory, iterative methods, cryptography and so on. So the expression for the group inverse (Drazin inverse) of the partitioned matrix is very important.In1979, Meyer proposed an open problem to find the representation for the Drazin inverse of the block matrix(?).In983, Campbell proposed an open problem to find an explicit representation of the Drazin inverse for the form of (?) when he researched the solution of differential equations.However, as the limitations of the methods and the difficultly of the problem, these problems have been not completely solved. However, there are many literatures about the representations of the group (Drazin) inverse for the block matrix (?) under some conditions.In Chapter1, we briefly give the development status and the significance for study the generalized inverses of matrices, and also give the basic knowledge of the generalized inverses theory in Chapter2. Finally, the main results of this paper are given in Chapter3and4, which are listed below:In the third chapter, we give the following results:(1) Representation of the group inverse of additive matrices P+Q is given, whereQP=0, P#and (PπQ)#exist.(2) Representation of the group inverse of additive matrices P+Q is given, wherePQP=0, QPπQ=0, QP#Q=PοQPπ.In the fourth chapter, (1)Representation of the group inverse of block matrix (?)∈K2n×2n is given, where A∈Kn×n,A#and(D-CA#B)#exist,CAπ=0,A.B=0,DSπC=0,BSπD=0, BSπC=0.(2)Representation of the group inverse of block matrix (?)∈K2n×2n is given, where A.D∈Kn×n, A.exists.D-CA.A is invertible.(3) The existence and representations of the group inverse of block matrix (?)∈K2n×2n is given,where A∈Kn×n,l1,l2,k1,k2, are positive integer number, A#exists,A1,A2∈{Al1(A#)k1,Al2(A#)k2(A#)k2}.(4)Representation of the group inverse of block matrix(?)∈K2n×2n is given, where A∈Kn×n,A#and B#exist,XAA#YB=0.