Representations Of The Group Inverse For Some Block Idempotent Matrices And Preserver Problems On Spaces Of Hermite Matrices

Generalized inverse of matrices theory is an active issue in the study of matrix theory.In 1979,Campbell and Meyer issued an open problem: finding explicit expressions for the Drazin inverse and group inverse of a 2×2block matrix (?) in terms of its various blocks,where the blocks,Aand D are assumed to be square matrices. This problem isn't solved yet up to now.Some scholars only give the expressions under some special circumstances. Linear Preserver Problems over matrix space is one of the most popular research topics in the international matrix theory field, which characterizes the problem of mapping preserving invariant over matrix space.Firstly,the status of generalized inverses of matrices and preserver problems about the generalized inverses were outlined,and the definition, quality of the generalized inverses , linear maps and the basic knowledge of the Hermite matrices are given in this paper. Finally, the expression of thegroup inverse of block matrices like (?) is discussed in the Chapter 5and also some expressions of group inverse of the block matrices and the new conclusions of its existence are given.In Chapter 6,this paper studies the Linear Maps Preserving of hermite matrices over real fields,characterizing the forms of its linear preserving operators.The following are main results:1.Giving the existences and the representations of the group inverse forblock matrices(?),(?),(?),(?),(?),where P∈C^n×n,P^k=P,P^* is the transpose conjugate of P.2.Giving the forms of linear maps preserving group inverses on hermite matrices space over real fields. The results of this paper have already been in press,which can be seen in enclosure.