分裂四元數矩陣幾類代數問題的算法研究
發(fā)布時間：2019-05-28 02:26
【摘要】：分裂四元數代數是結合代數,同時也是不可交換的四維Clifford代數,它包含零因子,冪零元和非平凡的冪等元.分裂四元數環(huán)和四元數環(huán)是兩種不同的非交換四維Clifford代數,后者是一個非交換的體,而前者不是.因此,分裂四元數環(huán)的代數結構比四元數環(huán)的代數結構更為復雜.當物理學家們在研究經典力學和Non-Hermitian量子力學的關系時,他們發(fā)現這些力學與四元數和分裂四元數有很大的聯(lián)系.這一發(fā)現使得運用四元數和分裂四元數的代數方法去解決經典力學中富有挑戰(zhàn)性的問題成為可能.本文主要對分裂四元數的代數方法問題進行研究,如分裂四元數的矩陣方程問題和矩陣的逆的問題和矩陣的秩的問題等.文章結構如下:第一章,主要介紹分裂四元數的代數方法在物理學中的應用的研究背景和發(fā)展現狀,以及本文的主要研究成果.第二章,探討分裂四元數矩陣是否可對角化的代數方法.通過在分裂四元數環(huán)中給出矩陣的兩種代數技巧,得到兩種代數方法使分裂四元數矩陣可實現對角化,最后通過算例驗證有效性.第三章,尋找代數方法解分裂四元數線性方程組.首先,給出異于問題一中的分裂四元數矩陣的另一種復表示方法.其次,結合分裂四元數矩陣的復表示矩陣定義分裂四元數矩陣的秩的概念,并得到了求解分裂四元數線性方程組的一種代數方法及相應算法.最后,用算例來說明此方法的可行性.第四章,研究用Cramer法則解分裂四元數線性方程組.首先,通過利用問題二中矩陣的代數技巧,自定義矩陣的行列式等概念并取得相應結果.然后,根據上面的討論,可得分裂四元數環(huán)上線性方程組的Cramer法則.最后,用算例來說明此方法的可行性.第五章,研究分裂四元數環(huán)中表示形式為方程(SQQP) x2 + bx+ c = 0的零點問題.按照在實數域上求解SQQP方程的理論,把分裂四元數環(huán)中SQQP方程化簡為一個帶二次約束條件的含未知數的實線性方程組,得到解SQQP的一種算法,給出算例來證明此方法是可行的.
[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.
【學位授予單位】：曲阜師范大學
【學位級別】：碩士
【學位授予年份】：2017
【分類號】：O151.21
本文編號：2486654
[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.
【學位授予單位】：曲阜師范大學
【學位級別】：碩士
【學位授予年份】：2017
【分類號】：O151.21
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