域上保持矩阵D-逆的线性算子

Linear operators preserving Drazin inverses of matrices over fields

设F是至少包含5个元素的域,令Mn(F)为F上的n×n全矩阵代数。在广义逆保持的研究中,特征为2的域上的工作尚不多见,并且由于工作难度大,关于特征2的情形的工作不仅没有加法映射的结果,而且即使是线性映射也只是讨论可逆的情形,并且在基础域附加一些条件。文中刻画当chF=2且n≥m≥2时,从Mn(F)到Mm(F)保持矩阵D-逆的线性算子的形式。利用保幂等的结论证明f为从Mn(F)到Mm(F)的保持矩阵D-逆的非零线性算子当且仅当存在P∈GLn(F),使得f(A)=PAP-1,A∈Mn(F);或者存在P∈GLn(F),使得f(A)=PAtP-1,A∈Mn(F)。

Let F be a field with at least five elements,Mn（F） the n×n full matrix algebra over F.However,when the characteristic of the base field was 2,as to the preserving of the generalized inverses,the results were less.As to the characteristic 2,because of higher difficulty no results on the addition maps were obtained and more the discussed linear maps were invertible plus more conditions on the base fields.In this paper,we determine the forms of linear maps from Mn（F） to Mm（F）preserving Drazin inverses of matrices under chF=2 and n≥m≥2.Using the conclusions of idempotent-preserving,it is proved that f is the nonzero linear maps from Mn（F） to Mm（F） preserving Drazin inverses of matrices,if and only if there exists P∈GLn（F）,such thatf（A）=PAP-1,A∈Mn（F）,or there exists P∈GLn（F）,such that f（A）=PAtP-1,A∈Mn（F）.