Research On Algorithms Based On The Eigenvalue Problem Of The Several Four-dimensional Algebras

Since the quaternion was proposed,scholars had continuously increased their research interest in using quaternions to solve problems.In order to meet the specific research needs of different fields,they had continued to explore and derived a variety of four-dimensional algebras from quaternions,including split quaternions,generalized quaternions,commutative quaternions,nectarines,conectarines,etc.For a long time,the eigenvalue problem of matrix is not only an important branch of numerical linear algebra,at the same time,it is also a basic problem in physics,thermodynamics and other technical aspects,it has a wide range of application value.It is also particularly important for the eigenvalue problem of four-dimensional algebra.So far,the eigenvalue problems of quaternion,split quaternion and commutative quaternion matrices had been discussed and studied successively,but according to the research of the author of this article,the results of the previous research on the eigenvalue problem of commutative quaternion matrices were not accurate enough,because the zero divisors of the commutative quaternions were ignored,etc.Therefore,this article will focus on the commutative quaternion algebra and the others unstudied nectarine algebra and v-quaternion algebra to study their matrix eigenvalue problems and algebraic methods,respectively.The content of the article mainly involves the following aspects:In the first part,briefly introduced the research status of several types of fourdimensional algebra in recent years,summarized and reviewed the research methods and completed results of scholars on the eigenvalue problems of four-dimensional algebraic matrices,so as to explain the research meaning and values on several types of four-dimensional algebraic eigenvalue problems.In the second part,according to Professor Soo-Chang Pei's research on the eigenvalue problems of commutative quaternion matrix,the concept of eigenvalue family of commutative quaternion matrices was introduced,which proved the incompleteness of the original theoretical results and gave new algebraic method and the corresponding research results,and compared with the original method to prove the rationality of the method in this article.In the third part,based on the eight four-dimensional algebras proposed by Professor B.Schmeikal,by introduced the real representation of the nectarine matrix,the nectarine matrix was isomorphically mapped to the real matrix for discussion,and the properties and equivalence classes of the nectarine algebra was introduced,and then studied the right eigenvalue problem of the nectarine matrix,and an algebraic method for finding the right eigenvalue and the corresponding eigenvector of the nectarine matrix was derived.In the fourth part,on the basis of the concept of v-quaternion,the complex representation of the v-quaternion matrix was introduced,and the algebraic technique for finding the right eigenvalues and corresponding eigenvectors of the v-quaternion matrix was derived.