Research On Several Algorithms Of The System Of Generalized Quaternionic Linear Equations

After the introduction of quaternions and split quaternions,the generalized quaternion was proposed by professor R.S.Pierce as a kind of associative algebra in 1982.Most scholars think that the generalized quaternion is a kind of generalization of the skew-field of quaternions.In 2014,professor B.Schmeikal introduced eight kinds of four-dimensional algebras and some properties.The algebras of quaternion(split quaternion)is also one of them.However,as far as the authors know,quaternion and split quaternion as two special cases of four-dimensional algebra.Scholars all have classified discussion the problems of the system of quaternion and split quaternion linear equations,and even have not given the related conclusions in the aspect of split quaternion.This paper aims to present,in a unified manner,algebraic techniques for linear equations which are valid on both the algebras of quaternion and split quaternion.Therefore,we focus on the algebras of v-quaternion and generalized quaternion,study the Cramer’s rule of the system of linear equations,the constructive methods for solving the system of linear equations and the least square problem of the system of linear equations.The main contents are as follows:In the first part,the research backgrounds and current situations of quaternion,split quaternion,generalized quaternion and other four-dimensional algebras are briefly introduced,and the research significances of the system of generalized quaternion linear equations and the main results of this paper are expounded;In the second part,the unified form of quaternion and split quaternion is introduced.The basic properties of algebra are discussed.The definitions of rank,adjoint matrix and inverse matrix of v-quaternion matrix are given by means of an isomorphic complex matrix representation of v-quaternion matrix,and the Cramer’s rule over v-quaternion algebra is derived;In the third part,on the basis of v-quaternion algebra,a new complex(real)matrix representation is introduced.By the relationship between the system of v-quaternion linear equations and corresponding complex(real)representation matrix equations,the constructive methods for solving the system of linear equation are derived and proved to be feasible;In the fourth part,based on the generalized quaternion algebra,the complex(real)matrix representation of generalized quaternion matrix and the definition of matrix norm are introduced.By the congruent relationship between the solution of the complex(real)representation equation and the system of original linear equations,two kinds of algebraic techniques for solving the least squares problem are derived.These two algebraic techniques are not only applicable to quaternion algebra and split quaternion algebra,they can also be applied to other four dimensional algebras(Nectarines,Conectarines).