四元数的实矩阵表示

Real Matrix Representation of Quaternions

四元数代数在计算机图形学、现代物理学、卫星的姿态表示等领域中都扮演着重要角色,从数学角度对四元数进行彻底研究是有价值的,但是,由于不可交换性,四元数并不像人们期望的那样易于掌握。处理四元数的方法之一是把它们等同为实数矩阵,其中各位置上的元素当然是可交换的。这样的"等同"实际上是从四元数代数到Rn×n的代数嵌入。研究了从四元数代数到Rn×n的代数嵌入问题,给出了把虚单位映成带符号置换矩阵条件下的所有可能的代数嵌入。我们用的方法是去考虑四元数代数生成元(即虚单位)的象。我们考察这些象的性质并确定有哪些实数矩阵满足它们。然后,我们运用群作用的语言化简了问题。我们得到的结论是有趣的:决定着嵌入的那些关键性的矩阵对是由的实数矩阵对构成的。而这样的矩阵对本质上只有两对。

Quaternion algebra plays an important role in computer graphics,modern physics,attitude representation of satellite among many others. It is valuable to make a thorough study of quaternion from a mathematical point of view. However,due to the non-commutative nature,quaternion algebra is not so easy to handle as people expected. One of the methods to deal with quaternions is to identify them with real matrices,the entries of which are of course commutative. These identifications are actually embeddings from quaternion algebra to Rn×n. The current method investigated the algebraic embedding problem from the quaternion algebra to Rn×n,and found out all the possible algebraic embeddings under the condition that maps the imaginary units to signed permutation matrices. The method is to consider the images of the generators （i.e. the imaginary units） of quaternion algebra.,then investigated the properties of these images and determined what kind of real matrices fulfilled them. Next,it solved the problem by invoking the language of group action. The conclusion turns out to be interesting： the crucial pairs of matrices,which determine the embeddings,are made up of real matrix pairs And there are essentially two pairs of the latter kind.