Real Matrix Representation of Quaternions

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.