拓展对称在正交变换中的应用

域上欧氏几何中，把正交变换表为对称之积的问题，是几何学中基本问题之一。70年代以后，环上几何学兴起，欧氏空间中把正交变换表为对称之积，为人们所注意。如何将这一问题的结果有效的转移到环上，转移过程中，出现一类对称叫拓展对称的问题。因此，欲将域上的结果有效地转到环上，首先必须解开拓展对称。在域上，开解正交变换表成对称之积，因子个数的多少，是用变换的剩余数来标定的。在环上，仅用剩余数却难于定出因子个数，于是创出了一个偏差数的概念。用正交变换的偏差数和剩余数字标定因子个数，表明分解的长度。同时论述了拓展对称在正交变换中的应用。

The Application of Extended Symmetry in Orthogonal Transformation

In field Euclidean geometry, the problem of expressing orthogonal transformation as symmetrical product is one of the basic problems in geometry. After 1970s, ring geometry sprang up in which orthogonal transformation is expressed as symmetrical product and it attracted much attention. A type of symmetry known as extended symmetry appears in the process of transferring the result of the problem to the ring. Therefore, the extended symmetry problem must be solved so as to transfer the result effectively to the ring. On the field, the number of factors in solving the symmetrical product is calibrated by the remainders of the transformation. On the ring, it is very difficult to determine the number of factors and the concept of deviation is created. This paper uses the deviation and remainder in orthogonal transformation to calibrate the number of factors and indicate the length of the resolution. Meanwhile, this paper discusses the application of extended symmetry in orthogonal transformation.