实四元数体上代数特征值分布与估计及奇异值分解研究
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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。
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.
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。
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.
引文
[1]孔国平,邱璐,王剑虹.数学史简编[M],科学出版社,1992,1-50.
[2]刘俊峰.三维转动的四元数表述[J],大学物理,2004(4),39-43.
[3]许方官,陆元荣.四元数在量子力学中的应用[J],大学物理,2001(11),20-23.
[4]吴小平,冯正平.四元数法在AUV六自由度仿真中的应用[J],船舶工程, 2007(1),9-11.
[5]宋文超,贾建援,王迎昆,陈贵敏.基于四元数方法的光学系统可视化建模与仿真[J],应用光学,Journal of applied optics, 2007 (2),174-177.
[6]叶佳,张建秋,胡波.客观评估彩色图像质量的超复数奇异值分解法[J],电子学报, Acta electronica sinica, 2007(1),28-32.
[7]刘海颖,王惠南,刘新文.基于UKF的四元数载体姿态确定[J],南京航空航天大学学报, 2006(1),37-42.
[8]崔峰,曹学光,彭思龙.新的四元数解析信号相位定义[J],中国图像图形学报,Journal of image and graphics, 2006(2), 107-114.
[9]韦娟,宁方立.误差四元数及其在航天器姿态控制中的应用[J],飞行力学, Flight dynamics, 2006(2),60-63.
[10]谢邦杰.Wedderburn定理及其推广与交换性问题[J],数学研究与论,1983(4),83-92.
[11]谢邦杰.环结构中的若干新成果(一)[J],数学研究与评论,1982(1),133-144.
[12]谢邦杰.关于交换性问题的近期发展概况[J],数学进展,1982(2), 113-124.
[13]谢邦杰.左零因子理想具升链条件之环[J],数学进展,1981(2),163-154.
[14]谢邦杰.体上特征矩阵的法式与弱法式存在定理[J],数学学报,1980(3),398-410.
[15]谢邦杰.自共轭四元数矩阵的行列式的展开定理及其应用[J],数学学报,1980(5),668-683.
[16]谢邦杰.体上矩阵的特征根与标准形式的应用[J],数学学报,1980(4),522-533.
[17]谢邦杰.四元数自共轭矩阵与行列式[J],吉林大学学报,理学版1980(2), 22-38.
[18]谢邦杰.任意体上可中心化矩阵的行列式[J],吉林大学学报,理学版,1980(3), 4-36.
[19]谢邦杰.环结构中的若干新成果(二) [J],数学研究与评论,1981(2),133-144.
[20]谢邦杰. Hadamard定理在四元数体上的推广[J],中国科学A辑, 1979(S1),23-29.
[21]谢邦杰.环与体上的矩阵及两类广义Jordan形式[J],吉林大学学报,理学版,1978(1),55-65.
[22]谢邦杰.体上矩阵的有理简化形式与Jordan形式的唯一性问题[J],吉林大学学报,理学版,1978(1),110-119.
[23]谢邦杰. Cochran定理在任意体上的推广[J],吉林大学学报理学版,1978(1), 120-121.
[24]谢邦杰.体上特征矩阵的简化形式的唯一性定理[J],吉林大学学报理学版,1979(1),110-119.
[25]屠伯埙.四元数体上自共轭阵的中心化基本定理及其应用[J],数学杂志,1988(2), 40-47.
[26]屠伯埙.四元数体上方阵非异性的判定[J],复旦学报,自然科学版,1988(2), 14-19.
[27]屠伯埙.四元数矩阵的左、右特征值的不等式[J],复旦学报,自然科学版,1992(4), 68-76.
[28]屠伯埙.亚正定阵理论(I),数学学报[J],1990(4), 32-41.
[29]屠伯埙.四元数体上矩阵的弱直积与弱圈积[J],复旦学报,自然科学版,1991(3), 94-103.
[30]屠伯埙.体上线性映射的子空间的维数及其应用[J],数学研究与评论,1990(3), 15-20.
[31]屠伯埙.体上逆阵的子阵的一个注记[J],数学研究与评论,1990(4), 113-117.
[32]屠伯埙.四元数矩阵的左、右特征值的不等式[J],复旦学报,自然科学版,1992(4), 68-76.
[33]屠伯埙.体上线性映射的子空间的维数及其应用[J],数学研究与评论,1990(3), 15-20.
[34]屠伯埙.亚正定阵理论(II)[J],数学学报,1991(1), 93-104.
[35]屠伯埙.除环上方阵的Dieudonné行列式[J],复旦学报,自然科学版,1990(1), 74-81.
[36]屠伯埙.四元数体上行列式上界估计的一般方法[J],复旦学报,自然科学版,1990(2), 33-39.
[37]屠伯埙.强p除环上方阵的酉相似理论(I)[J],数学研究与评论,1989(1), 15-25.
[38]屠伯埙.除环上矩阵的子矩阵的秩的恒等式与不等式[J],数学研究与评论,1989(3), 21-29.
[39]屠伯埙.正性除环上矩阵的正定自共轭分解[J],数学杂志,1989(3), 2-7.
[40]屠伯埙.四元数矩阵的UL分解,复旦学报,自然科学版[J],1989(20, 4-11.
[41]屠伯埙.强P除环上方阵的酉相似理论(Ⅱ),数学研究与评论[J],1988(1), 5-12.
[42]屠伯埙.两类迹占优阵的特征值的分布与估计,数学杂志[J],1988(1), 68-75.
[43]曾月迪,庄瓦金.关于四元数EP矩阵偏序的研究,数学研究[J],2007(1), 94-98.
[44]庄瓦金.四元数矩阵的极分解及其GL偏序,数学进展[J],2005(2), 61-67, 1-4.
[45]庄瓦金.四元数矩阵的Lowner偏序,数学物理学报[J],2004(5), 71-76.
[46]庄瓦金.半正定自共轭四元数矩阵广义逆的单调性问题,数学学报[J],1996(2), 33-39.
[47]庄瓦金.范畴中态射集减序的刻划,数学学报[J],1994(2), 172-179.
[48]庄瓦金.非交换主理想整环上分块矩阵的秩,数学研究与评论[J],1994(2), 265-270.
[49]庄瓦金.EP矩阵的Hartwig-Spindelbck问题与ρ.M-P逆的倒换顺序律[J],数学学报,1990(6), 73-79.
[50]庄瓦金.四元数矩阵的特征值与奇异值不等式,自然杂志[J],1989(4), 78-79.
[51]庄瓦金.四元数矩阵的加正定权的Moore-Penrose型广义逆的显公式[J],数学的实践与认识,1988(3), 70-76.
[52]庄瓦金.四元数矩阵的特征值与奇异值不等式,数学进展[J],1988(4),69-73.
[53]庄瓦金.范畴中态射的广义逆[J],数学学报,1988(1), 41-47.
[54]庄瓦金.四元数体上的矩阵方程,数学学报[J],1987(5), 114-120.
[55]庄瓦金.四元数矩阵的分解与Lavoie不等式的推广,数学研究与评论[J],1986(4), 25-27.
[56]庄瓦金.具有回复循环子块的块k-循环矩阵的某些性质[J],工程数学学报,1986(1),34-44.
[57]庄瓦金.体上矩阵的广义逆,数学杂志[J],1986(1), 107-114.
[58]陈龙玄,侯仁民,王亮涛.四元数矩阵的Jordan标准形,应用数学和力学[J],1996(6), 533-542.
[59]陈龙玄,侯仁民,王亮涛.Jordan canonical forms of matrices over quaternion field [J],应用数学和力学,英文版,1996(6), 559-568.
[60]陈龙玄.四元数体上的逆矩阵和重行列式的性质,中国科学A辑[J],1991(1), 25-35.
[61]陈龙玄. Cayley-Hamilton定理在四元数体上的推广,科学通报[J],1991(17), 13-15.
[62] A.Richard, G.Moshe. Stable norms on complex numbers and quaternions, Journal of algebra[J] 1999(219),1- 15.
[63] I.M. Michailov and N.P. Ziapkov, Embedding obstructions for the generalized quaternion group[J], Journal of algebra 2000(226), 375-389.
[64] L.Rowen, Y.Segevb. Normal subgroups generated by a single pure element in quaternion algebras[J], Journal of algebra 2006(305),130-145.
[65] O. Rojo, M. Soto. Bounds for sums of eigenvalues and applications[J], Computers and mathematics with applications ,2000 (39),1-15.
[66] Q.W.Wang. The general solution to a system of real quaternion matrix equation [J]. Comp. math. appL. 2005(49),665-675.
[67] Q.W. Wang. A system of four matrix equations over von Neumann regular rings and it applications. Acta, math, sinica , English series[J] (In Chinese)2005,(21:2), 323-334.
[68] Q.W. Wang. A system of matrix equation and a linear matrix equation over arbitrary regular rings with identity[J]. Linear algebra. appl. 2004(384), 43-54.
[69] L.P.Huang, W.So. On left eigenvalues of a quaternionic matrix[J], Linear algebra and its applications, 2001(323),105-116.
[70] L.P.Huang. Consimilarity of quaternion matrices and complex matrices, Linear algebra and its applications [J],2001(331), 21-30.
[71] L.P.Huang. On two questions about quaternion matrices[J], Linear algebra and its applications 2000(318),79-86.
[72] L.P.Huang. The extension of Roth’s theorem for matrix equations over a ring [J], Linear algebra and its applications,1997(259),229-235.
[73] L.P.Huang. The matrix equation AXB-GXD = E over the quaternion field[J], Linear algebra and its applications 1996(234), 197-205 .
[74]刘俊峰.三维转动的四元数表述[J],大学物理,2004(4), 40-44.
[75]许方官.四元数在量子力学中的应用[J],大学物理, 2001(11), 21-24.
[76]吴小平,冯正平,朱继懋.四元数法在AUV六自由度仿真中的应用[J],船舶工程,2007(1),9-11.
[77]宋文超,贾建援,王迎昆,陈贵敏.基于四元数方法的光学系统可视化建模与仿真[J],应用光学, Journal of applied optics,2007(2),58-61.
[78]叶佳,张建秋,胡波.客观评估彩色图像质量的超复数奇异值分解法[J],电子学报, Acta electronica sinica, 2007(1), 174-177.
[79]刘海颖,王惠南,刘新文.基于UKF的四元数载体姿态确定[J],南京航空航天大学学报, 2006(1), 37-42.
[80]崔峰,曹学光,彭思龙.新的四元数解析信号相位定义[J],中国图像图形学报,Journal of image and graphics, 2006(2), 107-114.
[81]韦娟,宁方立.误差四元数及其在航天器姿态控制中的应用[J],飞行力学, Flight dynamics, 2006(2), 60-63.
[82] J.M.Rico-Martinez and J. Gallardo-Alvarado, A simple method for the determination of angular velocity and acceleration of a spherical motion through quaternions[J], Mechanics. 2000(35), 111-118.
[83] J.Angeles. Rational Kinematics[M], Springer verlag, New York, 1988,33-56.
[84] A.Peres, Finite rotations and angular velocity [J], Am. J. phys. 1980 (48), 70–71.
[85] P.E.Nikravesh, R.A.Wehage, and O.K. Kwon, Euler parameters in computational kinematics and dynamics.Part 1[J], ASME J. Mech., Trans, Aut. Des. 1985(107), 358-365.
[86] P.E.Nikravesh, Computer-aided analysis of mechanical systems[J], Englewood cliffs, prentice hall,N.J,1988(9).235-245.
[87] A.A.Shabana, Dynamics of multibody systems, 2nd edition[M], Cambridge university press, cambridge, U.K.,1998,33-45.
[88] T.R. Kane and D.A. Levinson, Dynamics, theory and applications[M], Mcgraw hill, New York, 1985,123-135.
[89] J.M.Rico and J.Duffy, An application of screw algebra to the acceleration analysis of serial chains[J], Mech.mach. theory 1996 (31:4) ,445-457.
[90] E.T.Browne.The characteristic roots of a matrix[J]. Bull. Amer math soc 1930(36),705-710.
[91] R.S.Varga, Matrix iterative analysis[M]. Englewood cliffs, New jersey: prentice-hall, Inc1962,20-60.
[92] N.J.Pullman, Matrix theory and its application[M]. New York and Basel, Marcel Dekker, Inc, 1978, 11-66.
[93] K.Fan. Note on circular disks containing the eigenvalues of a matrix[J]. Duke math.1958(3),25-445.
[94] A. van der Sluis. Gerschgorin domains for partitioned matrices[J]. Linear algebra and itsapplication,1979(26),265-280.
[95] J.L.Wu. Distribution and estimation for eigenvalues of real quaternion matrices[J], Computers & mathematics with applications, 55 (2008) 1998–2004.
[96]屠伯埙.实四元数体上矩阵的弱直积与弱圈积[J].复旦学报(自然科学版),1991(3),337-338.
[97]庄瓦金.四元数矩阵特征值和奇异值不等式[J].数学进展, 1988(04),403-406.
[98] W.Zhang, T.Z.Huang, Y.X.Ying. A note on“disk theory for tensor product of matrix”[J]. Chinese journal of engineering mathematics, 2006(10), 912-914.
[99] W.T.Tong. Disk theory for tensor product of matrix [J]. The mathematics special issue of journal of Nanjing university, 1980, 92-107.
[100]屠伯埙.四元数体上的弱直积与弱圈积[J],复旦大学学报1991(3),331-339.
[101]庄瓦金.四元数体上的广义Lavoie不等式及其矩阵奇异值分解,数学研究与评1986(4),23-25.
[102]屠伯埙.关于“体上矩阵广义逆”一文的注[J],数学研究与评论,1986(4),26-26.
[103]孙建东.关于广义的正定矩阵型的进一步研究[J],高等学校计算数学学报.1996(1),94-96.
[104]沈光星.广义正定及其性质[J],高等学校计算数学学报.2002(2),186-192.
[105] C.R.Johnson, Positive definite matrices[J],Amer math monthly, 1970(77),259-264.
[106]夏长富.实方阵正定性的进一步推广[J],数学研究与评论, 1988(4),499-504.
[107]李炯生.实方阵的正定性[J],数学的实践与认识,1985(1),67-73.
[108]屠伯埙.四元数体上矩阵的弱直积与弱圈积[J],复旦大学学报(自然科学版),1991(3),331-339.
[109]吴雷.矩阵正定性的分块判定[J],科学通报. 1987(20),33-38.
[110]顾安娜.关于广义正定矩阵[J],数学的实践与认识,1994(2),48-50.
[111]杨新民.亚正定矩阵的行列式的一个不等式[J],数学的实践与认识,1994(2),45-47.
[112]王松桂,吴密霞,贾忠贞编著.矩阵不等式[M],科学出版社,2006,第2版,77-78.
[113]庄瓦金.体上矩阵理论导引[M].科学出版社,北京,2006, 124-135.
[114]屠伯埙.四元数体上矩阵的弱直积与弱圈积[J].复旦大学学报(自然科学版),1991(3),331-339.
[115] L.P.Huang.On left eigenvalues of a quaternionic matrix[J], Linear algebra and its applications, 2001(323), 105-116.
[116] F.Z.Zhang. Gerschgorin type theorems for quaternionic matrices [J], Linear algebra and its application. 2007(424), 139-153.
[117]伍俊良.自共轭四元数矩阵和的特征值积的不等式[J],重庆师范大学学报,1994(2), 91-93.
[118] J.L.Wu, L.M.Zou, X.P.Chen, S.J.Li. The estimation of eigenvalues of sum, difference, and tensor product of matrices over quaternion division algebra, Linear algebra and its application.428(2008),3023-3033.
[119]谢邦杰.自共轭四元数矩阵的行列式的展开定理及其应用[J] ,数学学报,1980(5), 678-683.
[120]陈龙玄.四元数体上的逆矩阵和重行列式的性质[J],中国科学(A辑),199(1),25-35.
[121]侯仁民,赵旭强,王亮涛.四元数体上的Vandermonde重行列式[J],应用数学和力,199(99), 100-107.
[122]张庆成.四元数体上重行列式的性质及其应用[J],数学学报,1995(2),253-259.
[123]李桃生.四元数体上重行列式的性质[J],华中师范大学学报(自然科学版) ,1995(1), 3-7.
[124]黄礼平.四元数矩阵的行列式及其性质[J],数学研究,1995(2),1-13.
[125]李文亮.四元数矩阵[M],国防科技大学,2002, 73-74.
[126]蒋尔雄.线性代数[M],人民教育出版社,1979(1),41-42.
[2]刘俊峰.三维转动的四元数表述[J],大学物理,2004(4),39-43.
[3]许方官,陆元荣.四元数在量子力学中的应用[J],大学物理,2001(11),20-23.
[4]吴小平,冯正平.四元数法在AUV六自由度仿真中的应用[J],船舶工程, 2007(1),9-11.
[5]宋文超,贾建援,王迎昆,陈贵敏.基于四元数方法的光学系统可视化建模与仿真[J],应用光学,Journal of applied optics, 2007 (2),174-177.
[6]叶佳,张建秋,胡波.客观评估彩色图像质量的超复数奇异值分解法[J],电子学报, Acta electronica sinica, 2007(1),28-32.
[7]刘海颖,王惠南,刘新文.基于UKF的四元数载体姿态确定[J],南京航空航天大学学报, 2006(1),37-42.
[8]崔峰,曹学光,彭思龙.新的四元数解析信号相位定义[J],中国图像图形学报,Journal of image and graphics, 2006(2), 107-114.
[9]韦娟,宁方立.误差四元数及其在航天器姿态控制中的应用[J],飞行力学, Flight dynamics, 2006(2),60-63.
[10]谢邦杰.Wedderburn定理及其推广与交换性问题[J],数学研究与论,1983(4),83-92.
[11]谢邦杰.环结构中的若干新成果(一)[J],数学研究与评论,1982(1),133-144.
[12]谢邦杰.关于交换性问题的近期发展概况[J],数学进展,1982(2), 113-124.
[13]谢邦杰.左零因子理想具升链条件之环[J],数学进展,1981(2),163-154.
[14]谢邦杰.体上特征矩阵的法式与弱法式存在定理[J],数学学报,1980(3),398-410.
[15]谢邦杰.自共轭四元数矩阵的行列式的展开定理及其应用[J],数学学报,1980(5),668-683.
[16]谢邦杰.体上矩阵的特征根与标准形式的应用[J],数学学报,1980(4),522-533.
[17]谢邦杰.四元数自共轭矩阵与行列式[J],吉林大学学报,理学版1980(2), 22-38.
[18]谢邦杰.任意体上可中心化矩阵的行列式[J],吉林大学学报,理学版,1980(3), 4-36.
[19]谢邦杰.环结构中的若干新成果(二) [J],数学研究与评论,1981(2),133-144.
[20]谢邦杰. Hadamard定理在四元数体上的推广[J],中国科学A辑, 1979(S1),23-29.
[21]谢邦杰.环与体上的矩阵及两类广义Jordan形式[J],吉林大学学报,理学版,1978(1),55-65.
[22]谢邦杰.体上矩阵的有理简化形式与Jordan形式的唯一性问题[J],吉林大学学报,理学版,1978(1),110-119.
[23]谢邦杰. Cochran定理在任意体上的推广[J],吉林大学学报理学版,1978(1), 120-121.
[24]谢邦杰.体上特征矩阵的简化形式的唯一性定理[J],吉林大学学报理学版,1979(1),110-119.
[25]屠伯埙.四元数体上自共轭阵的中心化基本定理及其应用[J],数学杂志,1988(2), 40-47.
[26]屠伯埙.四元数体上方阵非异性的判定[J],复旦学报,自然科学版,1988(2), 14-19.
[27]屠伯埙.四元数矩阵的左、右特征值的不等式[J],复旦学报,自然科学版,1992(4), 68-76.
[28]屠伯埙.亚正定阵理论(I),数学学报[J],1990(4), 32-41.
[29]屠伯埙.四元数体上矩阵的弱直积与弱圈积[J],复旦学报,自然科学版,1991(3), 94-103.
[30]屠伯埙.体上线性映射的子空间的维数及其应用[J],数学研究与评论,1990(3), 15-20.
[31]屠伯埙.体上逆阵的子阵的一个注记[J],数学研究与评论,1990(4), 113-117.
[32]屠伯埙.四元数矩阵的左、右特征值的不等式[J],复旦学报,自然科学版,1992(4), 68-76.
[33]屠伯埙.体上线性映射的子空间的维数及其应用[J],数学研究与评论,1990(3), 15-20.
[34]屠伯埙.亚正定阵理论(II)[J],数学学报,1991(1), 93-104.
[35]屠伯埙.除环上方阵的Dieudonné行列式[J],复旦学报,自然科学版,1990(1), 74-81.
[36]屠伯埙.四元数体上行列式上界估计的一般方法[J],复旦学报,自然科学版,1990(2), 33-39.
[37]屠伯埙.强p除环上方阵的酉相似理论(I)[J],数学研究与评论,1989(1), 15-25.
[38]屠伯埙.除环上矩阵的子矩阵的秩的恒等式与不等式[J],数学研究与评论,1989(3), 21-29.
[39]屠伯埙.正性除环上矩阵的正定自共轭分解[J],数学杂志,1989(3), 2-7.
[40]屠伯埙.四元数矩阵的UL分解,复旦学报,自然科学版[J],1989(20, 4-11.
[41]屠伯埙.强P除环上方阵的酉相似理论(Ⅱ),数学研究与评论[J],1988(1), 5-12.
[42]屠伯埙.两类迹占优阵的特征值的分布与估计,数学杂志[J],1988(1), 68-75.
[43]曾月迪,庄瓦金.关于四元数EP矩阵偏序的研究,数学研究[J],2007(1), 94-98.
[44]庄瓦金.四元数矩阵的极分解及其GL偏序,数学进展[J],2005(2), 61-67, 1-4.
[45]庄瓦金.四元数矩阵的Lowner偏序,数学物理学报[J],2004(5), 71-76.
[46]庄瓦金.半正定自共轭四元数矩阵广义逆的单调性问题,数学学报[J],1996(2), 33-39.
[47]庄瓦金.范畴中态射集减序的刻划,数学学报[J],1994(2), 172-179.
[48]庄瓦金.非交换主理想整环上分块矩阵的秩,数学研究与评论[J],1994(2), 265-270.
[49]庄瓦金.EP矩阵的Hartwig-Spindelbck问题与ρ.M-P逆的倒换顺序律[J],数学学报,1990(6), 73-79.
[50]庄瓦金.四元数矩阵的特征值与奇异值不等式,自然杂志[J],1989(4), 78-79.
[51]庄瓦金.四元数矩阵的加正定权的Moore-Penrose型广义逆的显公式[J],数学的实践与认识,1988(3), 70-76.
[52]庄瓦金.四元数矩阵的特征值与奇异值不等式,数学进展[J],1988(4),69-73.
[53]庄瓦金.范畴中态射的广义逆[J],数学学报,1988(1), 41-47.
[54]庄瓦金.四元数体上的矩阵方程,数学学报[J],1987(5), 114-120.
[55]庄瓦金.四元数矩阵的分解与Lavoie不等式的推广,数学研究与评论[J],1986(4), 25-27.
[56]庄瓦金.具有回复循环子块的块k-循环矩阵的某些性质[J],工程数学学报,1986(1),34-44.
[57]庄瓦金.体上矩阵的广义逆,数学杂志[J],1986(1), 107-114.
[58]陈龙玄,侯仁民,王亮涛.四元数矩阵的Jordan标准形,应用数学和力学[J],1996(6), 533-542.
[59]陈龙玄,侯仁民,王亮涛.Jordan canonical forms of matrices over quaternion field [J],应用数学和力学,英文版,1996(6), 559-568.
[60]陈龙玄.四元数体上的逆矩阵和重行列式的性质,中国科学A辑[J],1991(1), 25-35.
[61]陈龙玄. Cayley-Hamilton定理在四元数体上的推广,科学通报[J],1991(17), 13-15.
[62] A.Richard, G.Moshe. Stable norms on complex numbers and quaternions, Journal of algebra[J] 1999(219),1- 15.
[63] I.M. Michailov and N.P. Ziapkov, Embedding obstructions for the generalized quaternion group[J], Journal of algebra 2000(226), 375-389.
[64] L.Rowen, Y.Segevb. Normal subgroups generated by a single pure element in quaternion algebras[J], Journal of algebra 2006(305),130-145.
[65] O. Rojo, M. Soto. Bounds for sums of eigenvalues and applications[J], Computers and mathematics with applications ,2000 (39),1-15.
[66] Q.W.Wang. The general solution to a system of real quaternion matrix equation [J]. Comp. math. appL. 2005(49),665-675.
[67] Q.W. Wang. A system of four matrix equations over von Neumann regular rings and it applications. Acta, math, sinica , English series[J] (In Chinese)2005,(21:2), 323-334.
[68] Q.W. Wang. A system of matrix equation and a linear matrix equation over arbitrary regular rings with identity[J]. Linear algebra. appl. 2004(384), 43-54.
[69] L.P.Huang, W.So. On left eigenvalues of a quaternionic matrix[J], Linear algebra and its applications, 2001(323),105-116.
[70] L.P.Huang. Consimilarity of quaternion matrices and complex matrices, Linear algebra and its applications [J],2001(331), 21-30.
[71] L.P.Huang. On two questions about quaternion matrices[J], Linear algebra and its applications 2000(318),79-86.
[72] L.P.Huang. The extension of Roth’s theorem for matrix equations over a ring [J], Linear algebra and its applications,1997(259),229-235.
[73] L.P.Huang. The matrix equation AXB-GXD = E over the quaternion field[J], Linear algebra and its applications 1996(234), 197-205 .
[74]刘俊峰.三维转动的四元数表述[J],大学物理,2004(4), 40-44.
[75]许方官.四元数在量子力学中的应用[J],大学物理, 2001(11), 21-24.
[76]吴小平,冯正平,朱继懋.四元数法在AUV六自由度仿真中的应用[J],船舶工程,2007(1),9-11.
[77]宋文超,贾建援,王迎昆,陈贵敏.基于四元数方法的光学系统可视化建模与仿真[J],应用光学, Journal of applied optics,2007(2),58-61.
[78]叶佳,张建秋,胡波.客观评估彩色图像质量的超复数奇异值分解法[J],电子学报, Acta electronica sinica, 2007(1), 174-177.
[79]刘海颖,王惠南,刘新文.基于UKF的四元数载体姿态确定[J],南京航空航天大学学报, 2006(1), 37-42.
[80]崔峰,曹学光,彭思龙.新的四元数解析信号相位定义[J],中国图像图形学报,Journal of image and graphics, 2006(2), 107-114.
[81]韦娟,宁方立.误差四元数及其在航天器姿态控制中的应用[J],飞行力学, Flight dynamics, 2006(2), 60-63.
[82] J.M.Rico-Martinez and J. Gallardo-Alvarado, A simple method for the determination of angular velocity and acceleration of a spherical motion through quaternions[J], Mechanics. 2000(35), 111-118.
[83] J.Angeles. Rational Kinematics[M], Springer verlag, New York, 1988,33-56.
[84] A.Peres, Finite rotations and angular velocity [J], Am. J. phys. 1980 (48), 70–71.
[85] P.E.Nikravesh, R.A.Wehage, and O.K. Kwon, Euler parameters in computational kinematics and dynamics.Part 1[J], ASME J. Mech., Trans, Aut. Des. 1985(107), 358-365.
[86] P.E.Nikravesh, Computer-aided analysis of mechanical systems[J], Englewood cliffs, prentice hall,N.J,1988(9).235-245.
[87] A.A.Shabana, Dynamics of multibody systems, 2nd edition[M], Cambridge university press, cambridge, U.K.,1998,33-45.
[88] T.R. Kane and D.A. Levinson, Dynamics, theory and applications[M], Mcgraw hill, New York, 1985,123-135.
[89] J.M.Rico and J.Duffy, An application of screw algebra to the acceleration analysis of serial chains[J], Mech.mach. theory 1996 (31:4) ,445-457.
[90] E.T.Browne.The characteristic roots of a matrix[J]. Bull. Amer math soc 1930(36),705-710.
[91] R.S.Varga, Matrix iterative analysis[M]. Englewood cliffs, New jersey: prentice-hall, Inc1962,20-60.
[92] N.J.Pullman, Matrix theory and its application[M]. New York and Basel, Marcel Dekker, Inc, 1978, 11-66.
[93] K.Fan. Note on circular disks containing the eigenvalues of a matrix[J]. Duke math.1958(3),25-445.
[94] A. van der Sluis. Gerschgorin domains for partitioned matrices[J]. Linear algebra and itsapplication,1979(26),265-280.
[95] J.L.Wu. Distribution and estimation for eigenvalues of real quaternion matrices[J], Computers & mathematics with applications, 55 (2008) 1998–2004.
[96]屠伯埙.实四元数体上矩阵的弱直积与弱圈积[J].复旦学报(自然科学版),1991(3),337-338.
[97]庄瓦金.四元数矩阵特征值和奇异值不等式[J].数学进展, 1988(04),403-406.
[98] W.Zhang, T.Z.Huang, Y.X.Ying. A note on“disk theory for tensor product of matrix”[J]. Chinese journal of engineering mathematics, 2006(10), 912-914.
[99] W.T.Tong. Disk theory for tensor product of matrix [J]. The mathematics special issue of journal of Nanjing university, 1980, 92-107.
[100]屠伯埙.四元数体上的弱直积与弱圈积[J],复旦大学学报1991(3),331-339.
[101]庄瓦金.四元数体上的广义Lavoie不等式及其矩阵奇异值分解,数学研究与评1986(4),23-25.
[102]屠伯埙.关于“体上矩阵广义逆”一文的注[J],数学研究与评论,1986(4),26-26.
[103]孙建东.关于广义的正定矩阵型的进一步研究[J],高等学校计算数学学报.1996(1),94-96.
[104]沈光星.广义正定及其性质[J],高等学校计算数学学报.2002(2),186-192.
[105] C.R.Johnson, Positive definite matrices[J],Amer math monthly, 1970(77),259-264.
[106]夏长富.实方阵正定性的进一步推广[J],数学研究与评论, 1988(4),499-504.
[107]李炯生.实方阵的正定性[J],数学的实践与认识,1985(1),67-73.
[108]屠伯埙.四元数体上矩阵的弱直积与弱圈积[J],复旦大学学报(自然科学版),1991(3),331-339.
[109]吴雷.矩阵正定性的分块判定[J],科学通报. 1987(20),33-38.
[110]顾安娜.关于广义正定矩阵[J],数学的实践与认识,1994(2),48-50.
[111]杨新民.亚正定矩阵的行列式的一个不等式[J],数学的实践与认识,1994(2),45-47.
[112]王松桂,吴密霞,贾忠贞编著.矩阵不等式[M],科学出版社,2006,第2版,77-78.
[113]庄瓦金.体上矩阵理论导引[M].科学出版社,北京,2006, 124-135.
[114]屠伯埙.四元数体上矩阵的弱直积与弱圈积[J].复旦大学学报(自然科学版),1991(3),331-339.
[115] L.P.Huang.On left eigenvalues of a quaternionic matrix[J], Linear algebra and its applications, 2001(323), 105-116.
[116] F.Z.Zhang. Gerschgorin type theorems for quaternionic matrices [J], Linear algebra and its application. 2007(424), 139-153.
[117]伍俊良.自共轭四元数矩阵和的特征值积的不等式[J],重庆师范大学学报,1994(2), 91-93.
[118] J.L.Wu, L.M.Zou, X.P.Chen, S.J.Li. The estimation of eigenvalues of sum, difference, and tensor product of matrices over quaternion division algebra, Linear algebra and its application.428(2008),3023-3033.
[119]谢邦杰.自共轭四元数矩阵的行列式的展开定理及其应用[J] ,数学学报,1980(5), 678-683.
[120]陈龙玄.四元数体上的逆矩阵和重行列式的性质[J],中国科学(A辑),199(1),25-35.
[121]侯仁民,赵旭强,王亮涛.四元数体上的Vandermonde重行列式[J],应用数学和力,199(99), 100-107.
[122]张庆成.四元数体上重行列式的性质及其应用[J],数学学报,1995(2),253-259.
[123]李桃生.四元数体上重行列式的性质[J],华中师范大学学报(自然科学版) ,1995(1), 3-7.
[124]黄礼平.四元数矩阵的行列式及其性质[J],数学研究,1995(2),1-13.
[125]李文亮.四元数矩阵[M],国防科技大学,2002, 73-74.
[126]蒋尔雄.线性代数[M],人民教育出版社,1979(1),41-42.