A new algebra structure of quaternion
四元数的一种新的代数结构

In recent years, applications of quaternion matrices are getting more and more important and extensive in quantum mechanics and rigid mechanics. With the rapid development of the above disciplines, it becomes necessary for us to further study the theories of quaternion matrices, but the studies and applications are more difficult due to the non-commutation of quaternion,. In this paper, a new mathematical algebra method of quaternionic mechanics was presented. In the study of mathematical methods of quaternionic mechanics, a concept of companion vector was introduced and some properties of the companion vector were discussed. By the complex presentation of quaternion, this paper introduced new concepts of determinant and rank of quaternion matrices, studied some properties of determinant, rank, invertible matrix, characteristic value and characteristic vector and linear equations over quaternion field, and derived some simple algebraic methods of above subjects, and obtained simple Cramer's rule and Cayley-Hamilton theorem over quaternion field.