Preserving Group Inverses Of Symmtric Matrices Problems Over Fields

Preserving invariant problems over matrix space is an active research area in matrix theory. This paper investigated the linear preserver problems when the invariant was the generalized inverses of matrices. Let F be a field, M_n(F) the n×n full matrix algebra F over, and/denote the linear operator over M_n(F).The generalized inverses of matrices and the research status of preserver problems about the genernalized inverses were outlined. The definition, quality of the generalized inverses and the basic knowledge of the linear maps, and the basic knowledge of the group, ring, field were given in this paper.However, when the characteristic of the base field or the base ring was 2, as to the preserving of the genernalized inverses, the results were less. As to the characteristic 2, because of higher difficulty no results on the addition maps about preserving group inverses of symmetric matrice were obtained and more the discussed linear maps were invertible plus more conditions on the base fields. By the method of characterizing the images about bases of space, this paper removed the assumption that f was a linear bijection and gave the forms of linear operators from M_n(F) to itself preserving group inv- ersesof symmetric matrices when the characteristic of the field was 2 and prove ir with a new way.