M_n上保持部分等距的线性映射

Linear maps preserving partial isometries on Mn

目的主要对在Mn(C)保持部分等距的线性映射φ进行刻画。方法利用保秩性进行证明。结果与结论证明了φ保持部分等距的充分必要条件是存在酉矩阵U,V使得φ(X)=UXV,X∈Mn(C)或者φ(X)=UXtrV,X∈Mn(C)成立,其中Xtr表示矩阵X的转置。

Aim It is described a linear mapφon Mn（C） which preserves partial isometries.Methods The properties of preserving rank are used to prove the main theorem.Results and Conclusion It proves that φpreserves partial isometries if and only if there are unitary matrices U and V such that φ（X）=UXV,X∈Mn（C） or φ（X）=UXtrV,X∈Mn（C）,where Xtr denotes the transpose of a matrix X.