Linear Preserving D Inverses Of Matrices Problems Over Fields And {1} Inverses Over PID

Preserving invariant problems over matrix space is an activeresearch area in matrix theory. This paper investigated the linearpreserver problems when the invariant was the generalized inverses ofmatrices. Let F be a field, M_n(F) the n×n full matrix algebraover F, and f denote the linear operator over M_n(F)to M_n(F). Letbe Ra PID. M_n(R) the n×n full matrix algebra over R,and f denotethe linear bijection over M_n(R).The generalized inverses of matrices and the research status ofpreserver problems about the genernalized inverses were outlined. Thedefinition, quality of the generalized inverses and the basicknowledge of the linear maps were given in this paper.However, when the characteristic of the base field or the base ringwas 2, as to the preserving of the genernalized inverses, the resultswere less. As to the characteristic 2, because of higher difficultyno results on the addition maps were obtained and more the discussedlinear maps were invertible plus more conditions on the base fields.By the method of characterizing the images about bases of space,this paper gave the forms of linear operators from M_n(F) to M_n(F)preserving Drazin inverses of matrices when the characteristic of thefield was 2 and was not 2. And also the forms of linear operators fromM_n(R) to itself preserving commutative {1}-inverses of matrices werecharacterized when the characteristic of the field was not 2.