全矩阵空间保矩阵逆的加法映射

Additive maps preserving matric inverse between matric spaces over fields

设F是一个特征不为2及3的域,Mn(F)表示F上n×n矩阵全体,GLn(F)记F上一般线性群,N-1(F)表示从Mn(F)到Mm(F)的保矩阵逆的全部加法映射的集合。以矩阵逆作为不变量,研究不同矩阵空间上加法保持映射的形式,并采用直接刻画基底的矩阵逆保持算子形式的办法,刻画了N-1(F)中元素的形式。从结果可看出当n=2时的映射形式要比n≥3时的映射形式复杂得多。

Suppose F is a field of characteristic not 2 or 3, denote by Mn （F） the space of all n × n full matrices over F, by GLn（F） the general linear group on F. Let N-1 （F） be the set of all additive maps which preserves matrix inverse from Mn （F） to Mm （F） . For the invariant-matrix inverse, the forms of additive maps preserving matrix inverse between different matrix spaces are characterized. The form of element in the set N_ 1 （F） is described. From the result, it is seen that the case of n =2 is more complicated than the case of n≥3.