The Related Properties Of Quaternions And Split Quaternions

In this thesis,we mainly study the quaternion matrix,quaternion exponential function,quaternion logarithm function,solution of split quaternion linear equation,split quaternion sim-ilar and pseudosimilar.The thesis is divided into five chapters.In Chapter one,as an introduction,we mainly introduce the background of two kinds of quaternions,and put forward my own research directions.In Chapter two,combined with the current research results,the idea of using complex number matrices prove some important theorems of quaternion matrices.At the end of the chapter,it is proved that the necessary and sufficient condition that two different quaternions can be exchanged is that their imaginary part vectors are parallel.In Chapter three,based on the imaginary unit i,we propose the imaginary unit of quaternion u,and obtain the Euler formula of the quaternion:e^uθ=cosθ+u sinθ.Next,we generalize the exponential function from the complex field to quaternion field.In Chapter four,we mainly introduce the solution structure of split quaternion linear equa-tion,and obtain solutions of three kinds of linear equations of split quaternion.We propose the concepts of similarity and pseudosimilarity of split quaternion and the corresponding criterions.Two split quaternions a,b∈H_s-R are similar if and only if?（a）=?（b）,K（a）=K（b）.At the end of this chapter,we prove that the necessary and sufficient condition for the exchange of two different split quaternions is that their imaginary part vectors are parallel.In Chapter five,we mainly summarize the results of the research.