非奇异矩阵的仿射变换
The affine preservers of non-singular matrices
摘要：
当K是任意域时，研究了M_n（K）的仿射自同构使GL_n（K）稳定.利用狄厄多内关于奇异矩阵的极大仿射子空间的定理，这很容易归结为已知的当n>2或#K>2时的线性映射的情形。我们包括一个简短的新的证明更一般的弗兰德斯定理的仿射子空间的M_{p,q}（K）有界秩。我们还发现M_2（F_2）的使GL_2（F_2）稳定的仿射变换群并不完全由线性映射组成。利用F_2上的二次型理论，我们构造了它与辛群Sp_4（F_2）和对称群S_6之间的显式同构。
英文摘要：
When K is an arbitrary field, we study the affine automorphisms of M_n(K) that stabilize GL_n(K). Using a theorem of Dieudonné on maximal affine subspaces of singular matrices, this is easily reduced to the known case of linear preservers when n>2 or #K>2. We include a short new proof of the more general Flanders' theorem for affine subspaces of M_{p,q}(K) with bounded rank. We also find that the group of affine transformations of M_2(F_2) that stabilize GL_2(F_2) does not consist solely of linear maps. Using the theory of quadratic forms over F_2, we construct explicit isomorphisms between it, the symplectic group Sp_4(F_2) and the symmetric group S_6.
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