【作者】 伍俊良；

【导师】 李声杰；

【作者基本信息】 重庆大学 ， 计算数学， 2008， 博士

【摘要】 代数与数值代数是计算数学研究的两个重要方面,四元数体上代数是复数域代数的扩展。然而,由于四元数乘法的不可交换性,造成了它与复数域上的代数理论既有一定的联系,又有很大的差异,形成相对独立的内容体系,四元数代数问题涉及抽象的理论研究与具体的实践应用两个方面。近年来,四元数代数问题已经引起了数学和物理研究工作者的广泛兴趣,四元数体上代数的许多问题已经被研究,比如四元数体上的多项式,行列式,特征值和四元数代数方程组等。由于四元数乘法的不可交换性,造成了对四元数代数问题研究的困难。然而四元数代数理论变得日益重要。许多应用科学领域,比如物理学,图形图像识别,飞船姿态定位,3-D动画等等,人们开始使用四元数代数理论解决许多实际的问题。因此这使得人们需要对四元数代数理论作深入的研究。理论上,四元数体上许多代数问题需要人们研究和解决,比如四元数矩阵特征值的分布与估计问题,四元数多项式问题,四元数矩阵奇异值分解问题,四元数代数方程组的解的问题,四元数矩阵标准形和正定性问题等等,均需要深入的研究。本博士学位论文较为系统地分析了四元数体上一些重要的代数特征,论文通过在四元数体上建立四元数范数和广义球邻域概念,对四元数矩阵特征值的广义球邻域分布与定位、特征值球邻域的连通性和非连通性、特征值的最小球邻域包含问题,特征值的矩、实部与虚部的上下界估计进行了研究,获得了相应问题的一些定理。进而考虑到四元数乘法的不可交换性,对四元数矩阵的左特征值和右特征值的分布与估计进行了研究,得到一些有意义的结果。借助于自共轭四元数矩阵的性质,研究自共轭四元数矩阵和、差及张量积的特征值关系问题,获得了一些不等式定理。论文在推广了的Gerschgorin定理的基础上,解决了Cassini卵形定理在四元数体上的形式问题。论文研究了四元数体上两类特殊乘积即Kronecker乘积和Hadamard乘积的奇异值分解问题,获得一些奇异值分解定理和迹范数不等式。此外,论文将各种文献中研究的正定矩阵归结为四种类型,即Ⅰ型正定、Ⅱ型正定、Ⅲ型正定和Ⅳ型正定,并将相应类型正定矩阵的集合记为, S_Ⅰ(M), S_Ⅱ(M), S_Ⅲ(M) and S_Ⅳ(M) ,分别就复数域和四元数体上的四元数正定矩阵集合的关系进行了研究,获得四类正定矩阵的一些判定定理。最后,为了阐明四元数代数问题研究的广泛性,论文还研究了四元数体上的其它一些代数问题,如四元数正规矩阵的对角化问题,新型的Gerschgorin定理问题、新型的Ostrowski定理和新型的Brauer定理等,它们是对复数域上相关定理在四元数体上的推广与拓展。更多还原

【Abstract】 Algebra and numerical algebra are two important parts of computational mathematics. Quaternion algebra is the extension of algebra over complex field. However, because of the non-commutative multiplication of quaternion, it leads to two cases: There are not only some connections, but also vital differences between quaternion algebra and complex algebra, and it have formed an independent relatively algebra system. Quaternion algebra problems also involve two aspects: Not only does it have abstract theoretical research but also specific practice applications.In recent years, the algebra problem over quaternion division algebra has drawn the attention of researchers of mathematics and physics. Many problems of quaternion division algebra have been studied, such as polynomial, determinant, eigenvalue, and system of quaternion matrices equations and so on. It is not easy to study quaternion algebra problems because of the non-commutative multiplication of quaternions. However, quaternion algebra theory is getting more and more important. In many fields of applied science, such as physics, figure and pattern recognition, spacecraft attitude control, 3-D animation and so forth, people start making use of quaternion algebra theory to solve some actual problems. Therefore, it is necessary make a further study about quaternion algebra theory.Theoretically, many algebra problems on quaternion filed need studying and solving, such as distribution and estimation of real quaternion matrix eigenvalues, quaternion polynomial, and singular value decomposition of quaternion matrix, solution to the system of quaternion equations, canonical form and positive definite problems of quaternion matrix and so on.In this dissertation, some important algebra features are introduced systematically. The concepts of norm and generalized spherical neighborhood of quaternion are established, the location of quaternion matrix eigenvalues, connectivity of spherical neighborhood of eigenvalues, the smallest inclusion theorem and upper bound and lower bound estimation theorems for the moment, the real and imaginary part of the real quaternion matrices eigenvalues are studied. Furthermore, the location and estimation of quaternion matrices left and right eigenvalues are studied and some theorems are obtained. Some inequalities about eigenvalues of sum, difference and tensor product of matrixes on quaternion division algebra are also obtained. Based on the well-known Gerschgorin Theorem, the form of Cassini Theorem is solved on quaternion division algebra. Besides, theorems about singular value decomposition of Kronecker and Hadamard product of quaternion matrix are obtained.In this dissertation, the positive definite matrix is introduced in previous literatures which are divided into four types, namely,Ⅰ,Ⅱ,ⅢandⅣpositive definite matrices, of which the sets are denoted by S_Ⅰ(M), S_Ⅱ(M), S_Ⅲ(M) and S_Ⅳ(M) respectively and some judgment theorems about positive definite matrix are obtained. Finally, for the sake of state the variety of quaternion algebra problems, paper introduced some other conclusions about diagonalizable of normal matrix and location of algebra eigenvalues, such as the new Gerschgorin theorem, Ostrowski theorem and Brauer theorem, they are the generalization of related theorem on complex field.更多还原