分裂四元数矩阵几类代数问题的算法研究

发布时间：2019-05-28 02:26

【摘要】：分裂四元数代数是结合代数,同时也是不可交换的四维Clifford代数,它包含零因子,幂零元和非平凡的幂等元.分裂四元数环和四元数环是两种不同的非交换四维Clifford代数,后者是一个非交换的体,而前者不是.因此,分裂四元数环的代数结构比四元数环的代数结构更为复杂.当物理学家们在研究经典力学和Non-Hermitian量子力学的关系时,他们发现这些力学与四元数和分裂四元数有很大的联系.这一发现使得运用四元数和分裂四元数的代数方法去解决经典力学中富有挑战性的问题成为可能.本文主要对分裂四元数的代数方法问题进行研究,如分裂四元数的矩阵方程问题和矩阵的逆的问题和矩阵的秩的问题等.文章结构如下:第一章,主要介绍分裂四元数的代数方法在物理学中的应用的研究背景和发展现状,以及本文的主要研究成果.第二章,探讨分裂四元数矩阵是否可对角化的代数方法.通过在分裂四元数环中给出矩阵的两种代数技巧,得到两种代数方法使分裂四元数矩阵可实现对角化,最后通过算例验证有效性.第三章,寻找代数方法解分裂四元数线性方程组.首先,给出异于问题一中的分裂四元数矩阵的另一种复表示方法.其次,结合分裂四元数矩阵的复表示矩阵定义分裂四元数矩阵的秩的概念,并得到了求解分裂四元数线性方程组的一种代数方法及相应算法.最后,用算例来说明此方法的可行性.第四章,研究用Cramer法则解分裂四元数线性方程组.首先,通过利用问题二中矩阵的代数技巧,自定义矩阵的行列式等概念并取得相应结果.然后,根据上面的讨论,可得分裂四元数环上线性方程组的Cramer法则.最后,用算例来说明此方法的可行性.第五章,研究分裂四元数环中表示形式为方程(SQQP) x2 + bx+ c = 0的零点问题.按照在实数域上求解SQQP方程的理论,把分裂四元数环中SQQP方程化简为一个带二次约束条件的含未知数的实线性方程组,得到解SQQP的一种算法,给出算例来证明此方法是可行的.
[Abstract]:Split quaternion algebra is a associative algebra and an irreplaceable four-dimensional Clifford algebra, which contains zero factors, nilpotent elements and nontrivial idempotents. Split quaternion rings and quaternion rings are two different noncommutative four-dimensional Clifford algebra, the latter is a noncommutative body, but the former is not. Therefore, the algebra structure of split quaternion ring is more complex than that of quaternion ring. When physicists are studying the relationship between classical mechanics and Non-Hermitian quantum mechanics, they find that these mechanics are closely related to quaternions and split quaternions. This discovery makes it possible to use the algebra method of quaternion and split quaternion to solve the challenging problems in classical mechanics. In this paper, the algebra method of split quaternion is studied, such as the matrix equation of split quaternion, the inverse of matrix and the rank of matrix, and so on. The structure of this paper is as follows: in the first chapter, the research background and development status of the application of split quaternion algebra method in physics are introduced, as well as the main research results of this paper. In the second chapter, the algebra method of diagonalization of split quaternion matrix is discussed. By giving two kinds of algebra techniques of matrix in split quaternion ring, two kinds of algebra methods are obtained to make the split quaternion matrix diagonal. Finally, an example is given to verify the effectiveness of the split quaternion matrix. In chapter 3, we find the algebra method to solve the split quaternion linear equations. First of all, another complex representation method of split quaternion matrix is given, which is different from the split quaternion matrix in problem one. Secondly, the concept of rank of split quaternion matrix is defined by combining the complex representation matrix of split quaternion matrix, and an algebra method and corresponding algorithm for solving split quaternion linear equations are obtained. Finally, an example is given to illustrate the feasibility of this method. In chapter 4, the Cramer rule is used to solve the split quaternion linear equations. Firstly, by using the algebra technique of matrix in problem 2, the determinant of matrix is defined and the corresponding results are obtained. Then, according to the above discussion, the Cramer rule of linear equations over split quaternion rings can be obtained. Finally, an example is given to illustrate the feasibility of this method. In chapter 5, we study the zero problem of the equation (SQQP) x 2 bx c = 0 in the split quaternion ring. According to the theory of solving SQQP equation in real number domain, the SQQP equation in split quaternion ring is reduced to a real linear system with unknown numbers with quadratic constraints, and an algorithm for solving SQQP is obtained. an example is given to prove that the method is feasible.
【学位授予单位】：曲阜师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O151.21

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