The Addition Of The M-p Inverse To Maintain The Mapping

Suppose F is a field of characteristic not 2. Let n and m be two arbitrarypositive integers with n≥2. We denote by M_n(F) and S_n(F) the space of n×n fullmatrices and the space of n×n symmetric matrices over F, respectively. Recently,the preserver problem between different sets of matrices has been active, and somerelated references have shown that the preserver problem about M-P inverses ofmatrices between sets of matrices with different dimensions is still an open problem,so I study it basing on this problem in the paper. One of important techniques in thestudy of preserver problems is to reduce new preserver problems to the known ones,such as idempotence,rank one preserver and so on. In terms of the particularityand complication of M-P inverses of matrices, reducing the linear(additive) mapspreserving M-P inverses of matrices to the idempotent preserver, similar to thepreserver problem about other generalized inverses, is impractical even if under thecondition that the characteristic of F isn't 2, so I study the problem by searchingsome particular matrices directly in this paper. First, in the chapter 2, the formsof the linear maps from S_n(F) to M_m(F) preserving M-P inverses of matricesare characterized, and thereby the forms of the linear maps from S_n(F) to S_m(F)preserving M-P inverses of matrices are characterized by retricting the range ofimage of maps to S_m(F), and the forms of the linear maps from M_n(F) to M_m(F)preserving M-P inverses of matrices are given later. In chapter 3, the forms ofthe additive maps from M_n(F) to M_m(F) preserving M-P inverses of matrices aredescribed by using the linear results of chapter 2. As an application, the forms of thelinear maps from S_n(F) to M_m(F)(S_m(F)) and from M_n(F) to M_m(F) preserving{1,3}({1,4}) inverses of matrices and the additive maps from M_n(F) to M_m(F)preserving {1, 3}({1,4})inverses of matrices are also given.