欧氏空间的等积变换的性质

Properties of the Equi-Inner Product Transformation of Euclidean Space

首先给出了欧氏空间的等积变换的定义。其次给出4个引理并利用这些引理给出了有限维欧氏空间的两个线性变换为等积变换的充要条件,其中一个充要条件反应了两个等积变换在规范正交基下的矩阵关系,另一个充要条件反应了两个等积变换之间的关系。最后给出了无限维欧氏空间为等积变换的一个充要条件及等积变换的一个性质。

This Paper introduces the definition of the equi-inner transformation of Euclidean space. Then gives four lemmas and two necessary and sufficient conditions of what the linear transformation is the equi-inner transformation. One of the necessary and sufficient conditions hint the matrix relations between the two equi-inner transformation matrices under standard orthogonal basis, the other one hint the relationship between two equi-inner transformations. Finally, the necessary and sufficient condition and a property are derived for infinite dimension Euclidean space is an equi-inner transformation.