Research About The Properties Of Split Quaternions And Split Quaternion Matrices

The split quaternion is a generalization of complex numbers,which can be extended to higher dimensions and is an important part of Clifford algebra(geometric algebra).Split quaternions provide tools for solving the problems of numerical computation and spatial rotation in quantum mechanics,quantum field theory,spatial geometry,deep learn ing,physics,coding theory,signal processing and so on.The research about split quaternion and split quaternion matrix is a popular research content in Clifford algebra recently,which has not yet been studied maturely.When people study split quaternions and split quaternion matrices,they usually use their corresponding isomorphic matrix representation.Different researchers sometimes use different matrices to represent the same split quaternion but there is no article explaining the relationships between these different matrix representations.In view of the above problems,this thesis proposed a 2x2 real matrix representation of split quaternions,then studied the relationship between the given matrix representation and the matrix representation used in the references,and proved the isomorphic relation between different matrix representations.Based on this,we gave the 2m×2n real matrix representation of m×n split quaternion matrices and discussed the properties of split quaternion matrices systematically.Here are the concrete results:1.We proposed a 2×2 isomorphic real matrix representation of the split quaternion which has lower order and smaller calculation cost compared with the 2×2 complex matrix form or 4×4 real matrix form in previous references.Then we explored the inner relationships between the 2×2 matrix representation form we proposed and presented in the references,and proved the isomorphic relation between the different 2×2 matrix representation forms.2.Two application examples were given:two isomorphic 2m×2n real matrix forms of m×n split quaternion matrices and one isomorphic 2×2 complex matrix of quaternions.The former is beneficial to the study of split quaternion matrix properties,and the latter has a positive effect on further study of quaternions.3.When we want to study some properties of split quaternion matrices,we can turn to study the properties of the relative isomorphic real matrix,which can make the research more convenient.Analogous to the properties of real(complex)matrices,we discussed some definitions,operations and properties of split quaternion matrices.