保欧氏度量偏序的变换

Transformation with Keeping Euclidean Measurement Order

设L(R^n)表示n维欧氏空间R^n的所有线性变换构成的集合,‖ξ‖表示向量ξ的欧氏长度,由欧氏长度建立起向量间的序关系,令:PO(R^n)={f∈L(R^n)■|ξ,η∈R^(n×1),‖ξ‖≤‖η‖■‖f(ξ)‖≤‖f(η)‖}则PO(R^n)是欧氏空间R^n中保欧氏度量偏序变换构成的集合,讨论了PO(R^n)的结构,证明了保持这种序关系的变换由正交变换和伸缩变换组成.

Let L（R^n） be the linear transformation on n-dimensional Euclidean space R^n. Let ‖ξ‖ be the Euclidean length of vector ξ.According to the Euclid length of the vector, set up the order relation about the vector. Let PO（R^n）={f∈L（R^n）｜A↓ξ,η∈R^n×1,‖ξ‖≤‖η‖→‖f（ξ）‖≤‖f（η）‖} be the transformation with keeping Euclidean measurement order on Euclidean space R^n. In this paper the structure PO（R^n） is studied. It is shown the transformations with keeping the order relation are made up of the Orthogonal transformation and Stretching transformation.