域上保持对称矩阵群逆的线性算子

Linear maps preserving group inverse of symmetric matrices over field

设F是一个特征为2的域,|F|>4,令Mn(F),Sn(F),分别为全矩阵空间和对称矩阵空间.讨论了在特征为2的情况下从Sn(F)到Mn(F)上保持对称矩阵群逆的线性算子的表示形式问题.给出了在特征为2的情况下从Sn(F)到Sn(F)保持对称矩阵群逆的线性算子的表示形式.研究的保持问题不仅在数学理论上有着广泛研究,而且在系统控制、量子力学、微分几何、数理统计等领域有着广泛的实际应用背景.

Let F be a field of chF = 2, │F│〉 4, and Mn （ F）, Sn （F） be the spaces of all matrices and symmetric matrices respectively over F. In this paper we determine the forms of linear maps from S. （F） to M. （F） preserving group inverse of symmetric matrices under chF = 2 and also determine the forms of linear maps from S.（F） to itself preserving group inverses of symmetric matrices under chF = 2. The problem we are study not only be developed deeply in mathematic theory, but also be applied extensively on practice such as system control, quantum mechanics, differential geometry and mathematical statistics ,etc.