四元数体上代数的若干矩阵问题研究
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摘要
四元数是继复数后又一新的数系,四元数体上代数是复数域上代数的扩展。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,又有很大的差别,形成相对独立的内容体系。
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。四元数体上代数问题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。然而,四元数体上许多代数问题还需要人们进行进一步的研究,比如四元数矩阵特征值的估计与对角化、四元数矩阵的广义特征值分布与估计、四元数右线性方程组解的扰动性估计问题、四元数矩阵的次亚正定性问题、四元数矩阵方程的可解性问题等等。本文较为系统地分析了四元数体上一些重要的代数特征,主要内容和创新点包括:
1.在四元数体上根据特征值的基本概念,将复数域上著名的Gerschgorin圆盘定理推广到四元数体上。由于四元数乘法的不可交换性,得到两种形式的四元数矩阵特征值分布定理,研究了四元数体上严格对角占优矩阵特征值的一些性质。同时给出了四元数矩阵广义特征值的定义,讨论了四元数矩阵左右广义特征值的性质,得到四元数正则矩阵束的广义特征值为实数的结论。获得了估计四元数矩阵广义特征值的Gerschgorin型定理,利用广义瑞利商这一有效的工具,获得了四元数矩阵广义特征值的上下界估计定理。
2.对四元数矩阵的对角化进行研究,获得了四元数矩阵可对角化的充要条件,并指出了四元数矩阵的对角化与实(复)数域上矩阵对角化的区别,说明了四元数体上的矩阵性质与实(复)数域上矩阵性质的差异。
3.本文在谱半径概念的基础上,讨论了谱半径的估计。
4.借助于四元数向量和四元数矩阵的范数理论解决了四元数矩阵求逆、线性方程组的误差估计问题。
5.对于次对角线方向上的情形,即四元数体上次亚正定矩阵,本文也作了一些研究,得到一些重要结果。
6.矩阵的Kronecker积是一种重要的矩阵乘积,由于四元数乘法的不可交换性,因此,四元数矩阵的Kronecker积性质有所不同于实(复)数矩阵的Kronecker积性质。利用四元数矩阵的Kronecker积这一有效的工具,研究了Lyapunov四元数矩阵方程与Stein四元数矩阵方程的可解性问题。
Quaternion is another new number system after the complex. The quaternion field is an extension of complex field. It is a relative independent system and has great difference with the complex field because of the non- commutative multiplication of quaternions.
In recent years, the algebra problems over quaternion division algebra have drawn the attention of researchers of mathematics and physics. Many problems of quaternion division algebra have been studied, such as polynomial, determinant, eigenvalues, and system of quaternion matrices equations and so on.
However, there are many questions to be studied in quaternion algebra, such as the estimation of quaternion matrices eigenvalues, diagonalization, the distribution for the generalized eigenvalues of quaternion matrices, the perturbation of solutions of quaternion right linear equations, the sub- positive of quaternion matrices and the solutions of quaternion matrix equations, e.t.c.The purpose of this paper is to discuss some important algebra properties on quaternion division algebra. The main results and innovations are listed as the following:
1. The famous Gerschgorin’s disk theorem over complex field is generalized to real quaternion field by the definition of eigenvalues. Two different forms of distribution of the left eigenvalues and right eigenvalues are obtained and some properties of eigenvalues of quaternion diagonally dominant matrices are also discussed. Moreover, the definition of generalized eigenvalues and its properties are given, and then we get a conclusion that the generalized eigenvalue of regular matrix is real. The estimation of the upper and lower bound of quaternion matrices’generalized eigenvalues is obtained with the guidance of the generalized Rayleigh quotient.
2. The necessary and sufficient condition for the diagonalized quaternion matrices is given, which is different from real (complex) field. Meanwhile, the difference between diagonalization of complex matrices and quaternion matrices is shown.
3. The estimation of spectral radius is discussed based on its concept.
4. We solve the quaternion matrices inversion, the estimation of linear equations error and eigenvalues according to quaternion vector and matrices norm theory.
5. We also study on the metapositive definite matrices over quaternion division algebra and some results are obtained.
6. The Kronecker product is important. The quaternion field is different from the complex field because of the non- commutative multiplication of quaternions. Then we study the solutions of quaternion Lyapunov matrix equations and the Stein Equations by Kronecker product over quaternion division algebra.
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。四元数体上代数问题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。然而,四元数体上许多代数问题还需要人们进行进一步的研究,比如四元数矩阵特征值的估计与对角化、四元数矩阵的广义特征值分布与估计、四元数右线性方程组解的扰动性估计问题、四元数矩阵的次亚正定性问题、四元数矩阵方程的可解性问题等等。本文较为系统地分析了四元数体上一些重要的代数特征,主要内容和创新点包括:
1.在四元数体上根据特征值的基本概念,将复数域上著名的Gerschgorin圆盘定理推广到四元数体上。由于四元数乘法的不可交换性,得到两种形式的四元数矩阵特征值分布定理,研究了四元数体上严格对角占优矩阵特征值的一些性质。同时给出了四元数矩阵广义特征值的定义,讨论了四元数矩阵左右广义特征值的性质,得到四元数正则矩阵束的广义特征值为实数的结论。获得了估计四元数矩阵广义特征值的Gerschgorin型定理,利用广义瑞利商这一有效的工具,获得了四元数矩阵广义特征值的上下界估计定理。
2.对四元数矩阵的对角化进行研究,获得了四元数矩阵可对角化的充要条件,并指出了四元数矩阵的对角化与实(复)数域上矩阵对角化的区别,说明了四元数体上的矩阵性质与实(复)数域上矩阵性质的差异。
3.本文在谱半径概念的基础上,讨论了谱半径的估计。
4.借助于四元数向量和四元数矩阵的范数理论解决了四元数矩阵求逆、线性方程组的误差估计问题。
5.对于次对角线方向上的情形,即四元数体上次亚正定矩阵,本文也作了一些研究,得到一些重要结果。
6.矩阵的Kronecker积是一种重要的矩阵乘积,由于四元数乘法的不可交换性,因此,四元数矩阵的Kronecker积性质有所不同于实(复)数矩阵的Kronecker积性质。利用四元数矩阵的Kronecker积这一有效的工具,研究了Lyapunov四元数矩阵方程与Stein四元数矩阵方程的可解性问题。
Quaternion is another new number system after the complex. The quaternion field is an extension of complex field. It is a relative independent system and has great difference with the complex field because of the non- commutative multiplication of quaternions.
In recent years, the algebra problems over quaternion division algebra have drawn the attention of researchers of mathematics and physics. Many problems of quaternion division algebra have been studied, such as polynomial, determinant, eigenvalues, and system of quaternion matrices equations and so on.
However, there are many questions to be studied in quaternion algebra, such as the estimation of quaternion matrices eigenvalues, diagonalization, the distribution for the generalized eigenvalues of quaternion matrices, the perturbation of solutions of quaternion right linear equations, the sub- positive of quaternion matrices and the solutions of quaternion matrix equations, e.t.c.The purpose of this paper is to discuss some important algebra properties on quaternion division algebra. The main results and innovations are listed as the following:
1. The famous Gerschgorin’s disk theorem over complex field is generalized to real quaternion field by the definition of eigenvalues. Two different forms of distribution of the left eigenvalues and right eigenvalues are obtained and some properties of eigenvalues of quaternion diagonally dominant matrices are also discussed. Moreover, the definition of generalized eigenvalues and its properties are given, and then we get a conclusion that the generalized eigenvalue of regular matrix is real. The estimation of the upper and lower bound of quaternion matrices’generalized eigenvalues is obtained with the guidance of the generalized Rayleigh quotient.
2. The necessary and sufficient condition for the diagonalized quaternion matrices is given, which is different from real (complex) field. Meanwhile, the difference between diagonalization of complex matrices and quaternion matrices is shown.
3. The estimation of spectral radius is discussed based on its concept.
4. We solve the quaternion matrices inversion, the estimation of linear equations error and eigenvalues according to quaternion vector and matrices norm theory.
5. We also study on the metapositive definite matrices over quaternion division algebra and some results are obtained.
6. The Kronecker product is important. The quaternion field is different from the complex field because of the non- commutative multiplication of quaternions. Then we study the solutions of quaternion Lyapunov matrix equations and the Stein Equations by Kronecker product over quaternion division algebra.
引文
[1]华罗庚,万哲先.典型群[M].上海:上海科学技术出版社,1963.
[2]谢邦杰.任意体上可中心化矩阵的行列式[J].吉林大学自然科学学报,1980,3:1-33.
[3]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,1980,23(5):668-683.
[4] Xiaoshan Chen, Wen Li. A note on the perturbation bounds of eigenspaces for Hermitian matrices[J]. J. Comput. Appl. Math.2006, 196:338-346.
[5]谢邦杰.关于自共轭四元数矩阵与行列式的几个定理[J].吉林大学自然科学学报,1982,1;1-7.
[6]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),1991,1:23-33.
[7]陈龙玄,侯仁民,王亮涛.四元数矩阵的Jordan标准形[J].应用数学和力学,1996,17(6):533-542.
[8] J.M. Varah. A lower bound for the smallest singular value [J]. Linear Algebra Appl. 1975, 11 : 3–5.
[9]陈龙玄.四元数矩阵的特征值和特征向量[J].烟台大学学报(自然科学与工程版),1993,3:1-8.
[10]李桃生.四元数矩阵的特征值与特征向量[J].华中师范大学学报(自然科学版),1995,12:407-411.
[11]黄礼平.四元数矩阵的特征值与奇异值估计[J].数学研究与评论,1992,3:449-454.
[12]郭时光.四元数方阵的正定性[J].四川轻化工学院学报,1999,1:35-38.
[13] Wu Junliang, Zou Limin, etc. The estimation of eigenvalues of sum, difference, and tensor product of matrices over quaternion division algebra[J].Linear Algebra Appl, 2008, 428: 3023–3033.
[14]李文亮.四元数矩阵奇异值的一个不等式[J].贵州大学学报(自然科学版),1993,10(3):154-156.
[15] Fuzhen Zhang. Gerschgorin type theorems for quaternionic matrices [J]. Linear Algebra Appl. 2007, 424:139– 153.
[16]李文亮.四元数自共轭矩阵迹的一些不等式[J].长沙电力学院学报(自然科学版),1996,11(1):5-11.
[17]张光枢.刚体有限转动合成的可交换性[J].力学学报,1982,4:34-38.
[18] Limin Zou. Simultaneous diagonalization of two quaternion normal matrices[J].Taiyuan. Word Academic Press. England.2008:481-484.
[19]李文亮.四元数矩阵[M].湖南:国防科技大学出版社, 2002.
[20]庄瓦金.体上矩阵理论导引[M].北京:科学出版社,2006.
[21] Yaotang Li, Fubin Chen, Deng Wang. New lower bounds on eigenvalue of the Hadamard product of an M-matrix and its inverse[J]. Linear Algebra Appl. 2009, 430:1423–1431.
[22]谢邦杰.体上矩阵的特征根与标准形式的应用[J].数学学报,1980,23(4):522-533.
[23]谢邦杰.环与体上矩阵及其两类广义Jordan形式[J].吉林大学自然科学学报,1978,1:21-46.
[24]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].吉林大学自然科学学报,1978,1:107-116.
[25]谢邦杰.体上特征矩阵的简化形式的唯一性问题[J].吉林大学自然科学学报,1979,1:1-10.
[26] Erxiong Jiang. An upper bound for the spectral condition number of a diagonalizable matrix [J]. Linear Algebra Appl. 1997, 262: 165–178.
[27]张远达.线性代数原理[M].上海:上海教育出版社: 1984:457-461.
[28]黄礼平.具有奇异值分解的代数[J].数学学报,1997,40(2):161-166.
[29]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,1988,8(2):143-150.
[30] Liping Huang, Wasin So. On left eigenvalues of a quaternionic matrix [J]. Linear Algebra Appl. 2001, 323:105– 116.
[31]曹重光.四元数自共轭矩阵的几个定理[J].数学研究与评论,1988,8:346-348.
[32]杨忠鹏.四元数矩阵乘积的奇异值与特征值不等式[J].数学研究与评论,1992,12:617-622.
[33]古以熹.矩阵特征值的分布[J].应用数学学报,1994,17(4):501-511.
[34] M.Fiedler, T.L. Markham. An inequality for the Hadamard product of M matrix and an inverse M matrix [J]. LinearAlgebra Appl. 1988, 102:1-8.
[35]庄瓦金.四元数矩阵的分解与Vavoie不等式的推广[J].数学研究与评论,1986,4:23—25.
[36] Tingzhu Huang. A note on generalized diagonally dominant matrices [J]. Linear Algebra Appl. 1995, 225:237–242.
[37]田永革.关于四元数矩阵可对角化的一个结果[J].河北师范大学学报(自然科学版),1993.1:21-23.
[38] R.A.Jacobs. Increased Rates of Convergence through Learning Rate Adaptation [J]. Neural Networks, 1988, 1 (4):295-307.
[39]屠伯埙.除环上方阵的Dieudonné行列式[J].复旦学报(自然科学版),1990,29(1):71-78.
[40] X.G.Wang, Z.Tang, H.Tamura. A modified error function for the back-propagation algorithm [J]. Neurocomputing, 2004, 57:477-484.
[41]姜同松,陈丽.四元数体上矩阵的广义对角化[J].应用数学和力学,1999,20(11):1203-1210.
[42] Qingwen 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.
[43]刘裔宏.关于广义特征值估计的一个Gerschgorin型定理[J].工科数学, 1994,10:49- 51.
[44]李仁仓.广义特征值的扰动界限[J].计算数学,1989,5(2):196-204.
[45] Jiang T S, Wei M S. Equality constrained least squares problem over quaternion field [J]. Applied Mathematics Letters, 2003, 16:883—888.
[46]屠伯埙.Schur定理在四元数体上的推广[J].数学年刊.1988,9A(2):130-138.
[47]姜同松,魏木生.四元数矩阵的实表示与四元数矩阵方程[J].数学物理学报,2006,20A(4):578-584.
[48] Rong Huang. Some inequalities for the Hadamard product and the Fan product of matrices [J]. Linear Algebra Appl. 2008, 428:1551–1558.
[49]杨忠鹏.关于四元数体上行列式的注记[J].新疆大学学报(自然科学版),1997,14(2):35-39.
[50] Johnson C.R. Positive definite matrices[J].Amer Math Monthly, 1970,77(1):259-264.
[51] Erxiong Jiang, Xingzhi Zhan. Lower bounds for the spread of a Hermitian matrix [J]. Linear Algebra Appl. 1997, 256: 153–163.
[52] Ning Long, Fengli Zhang. Novel Newton’s learning algorithm of neural networks [J]. Journal of Systems Engineering and Electronics, 2006, 17 (2): 450-454.
[53] A. Berman, R.J. Plemmons. Nonnegative Matrices in the Mathematical Sciences [M]. Academic Press, New York, 1979.
[54] D.Z.Dokovic, B.H.Smith. Quaternionic matrices: Unitary similarity, simultaneous traingula- rization and some trace identities [J]. Linear Algebra Appl. 2008, 428:890-910.
[55]屠伯埙.亚正定阵理论(I) [J].数学学报,1990,33(4):462-471.
[56]庄瓦金.四元数体上的矩阵方程[J].数学学报,1987,30(5):688-694.
[57] S.L. Adler. Quaternionic Quantum Mechanics and Quantum Fields [M]. Oxford University Press, New York, 1995.
[58]王卿文,薛有才.体与环上的矩阵方程[M].北京:知识出版社,1996.
[59] Liping Huang, Zhuojun Liu. Similarity reduction of matrix over a quaternion division ring [J]. Linear Algebra Appl. 2008, 428:765-780.
[60]伍俊良.四元数体上矩阵的弱直积与弱圈积德特征值和奇异值不等式[J].重庆师范学院学报(自然科学版),1994,11(4):60-65.
[61] P.M. Cohn. Skew Field Constructions [M]. London Mathematical Society Lecture Note Series,Cambridge University Press, Cambridge, 1977.
[62] F. Zhang. Quaternions and matrices of quaternions [J]. Linear Algebra Appl. 1997,251(1):21–57.
[63] L.P. Huang. On two questions about quaternion matrices [J]. Linear Algebra Appl. 2000 ,318(2):79–86.
[64] Q.W.Wang, The General Solution to a system of real quaternion matrix equation [J]. Computer and Mathematics with Applications .2005,49(1):665-675.
[65] Q.W. Wang. A system of four matrix equations over von Neumann regular rings and it applications [J]. Acta, Math, Sinica , English series .2005,21(4):323-334.
[66] Huang Liping.The matrix equation AXB-CXD=E over the quaternion field[J]. Linear Algebra Appl. 1996,234(2):197-208.
[2]谢邦杰.任意体上可中心化矩阵的行列式[J].吉林大学自然科学学报,1980,3:1-33.
[3]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,1980,23(5):668-683.
[4] Xiaoshan Chen, Wen Li. A note on the perturbation bounds of eigenspaces for Hermitian matrices[J]. J. Comput. Appl. Math.2006, 196:338-346.
[5]谢邦杰.关于自共轭四元数矩阵与行列式的几个定理[J].吉林大学自然科学学报,1982,1;1-7.
[6]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),1991,1:23-33.
[7]陈龙玄,侯仁民,王亮涛.四元数矩阵的Jordan标准形[J].应用数学和力学,1996,17(6):533-542.
[8] J.M. Varah. A lower bound for the smallest singular value [J]. Linear Algebra Appl. 1975, 11 : 3–5.
[9]陈龙玄.四元数矩阵的特征值和特征向量[J].烟台大学学报(自然科学与工程版),1993,3:1-8.
[10]李桃生.四元数矩阵的特征值与特征向量[J].华中师范大学学报(自然科学版),1995,12:407-411.
[11]黄礼平.四元数矩阵的特征值与奇异值估计[J].数学研究与评论,1992,3:449-454.
[12]郭时光.四元数方阵的正定性[J].四川轻化工学院学报,1999,1:35-38.
[13] Wu Junliang, Zou Limin, etc. The estimation of eigenvalues of sum, difference, and tensor product of matrices over quaternion division algebra[J].Linear Algebra Appl, 2008, 428: 3023–3033.
[14]李文亮.四元数矩阵奇异值的一个不等式[J].贵州大学学报(自然科学版),1993,10(3):154-156.
[15] Fuzhen Zhang. Gerschgorin type theorems for quaternionic matrices [J]. Linear Algebra Appl. 2007, 424:139– 153.
[16]李文亮.四元数自共轭矩阵迹的一些不等式[J].长沙电力学院学报(自然科学版),1996,11(1):5-11.
[17]张光枢.刚体有限转动合成的可交换性[J].力学学报,1982,4:34-38.
[18] Limin Zou. Simultaneous diagonalization of two quaternion normal matrices[J].Taiyuan. Word Academic Press. England.2008:481-484.
[19]李文亮.四元数矩阵[M].湖南:国防科技大学出版社, 2002.
[20]庄瓦金.体上矩阵理论导引[M].北京:科学出版社,2006.
[21] Yaotang Li, Fubin Chen, Deng Wang. New lower bounds on eigenvalue of the Hadamard product of an M-matrix and its inverse[J]. Linear Algebra Appl. 2009, 430:1423–1431.
[22]谢邦杰.体上矩阵的特征根与标准形式的应用[J].数学学报,1980,23(4):522-533.
[23]谢邦杰.环与体上矩阵及其两类广义Jordan形式[J].吉林大学自然科学学报,1978,1:21-46.
[24]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].吉林大学自然科学学报,1978,1:107-116.
[25]谢邦杰.体上特征矩阵的简化形式的唯一性问题[J].吉林大学自然科学学报,1979,1:1-10.
[26] Erxiong Jiang. An upper bound for the spectral condition number of a diagonalizable matrix [J]. Linear Algebra Appl. 1997, 262: 165–178.
[27]张远达.线性代数原理[M].上海:上海教育出版社: 1984:457-461.
[28]黄礼平.具有奇异值分解的代数[J].数学学报,1997,40(2):161-166.
[29]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,1988,8(2):143-150.
[30] Liping Huang, Wasin So. On left eigenvalues of a quaternionic matrix [J]. Linear Algebra Appl. 2001, 323:105– 116.
[31]曹重光.四元数自共轭矩阵的几个定理[J].数学研究与评论,1988,8:346-348.
[32]杨忠鹏.四元数矩阵乘积的奇异值与特征值不等式[J].数学研究与评论,1992,12:617-622.
[33]古以熹.矩阵特征值的分布[J].应用数学学报,1994,17(4):501-511.
[34] M.Fiedler, T.L. Markham. An inequality for the Hadamard product of M matrix and an inverse M matrix [J]. LinearAlgebra Appl. 1988, 102:1-8.
[35]庄瓦金.四元数矩阵的分解与Vavoie不等式的推广[J].数学研究与评论,1986,4:23—25.
[36] Tingzhu Huang. A note on generalized diagonally dominant matrices [J]. Linear Algebra Appl. 1995, 225:237–242.
[37]田永革.关于四元数矩阵可对角化的一个结果[J].河北师范大学学报(自然科学版),1993.1:21-23.
[38] R.A.Jacobs. Increased Rates of Convergence through Learning Rate Adaptation [J]. Neural Networks, 1988, 1 (4):295-307.
[39]屠伯埙.除环上方阵的Dieudonné行列式[J].复旦学报(自然科学版),1990,29(1):71-78.
[40] X.G.Wang, Z.Tang, H.Tamura. A modified error function for the back-propagation algorithm [J]. Neurocomputing, 2004, 57:477-484.
[41]姜同松,陈丽.四元数体上矩阵的广义对角化[J].应用数学和力学,1999,20(11):1203-1210.
[42] Qingwen 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.
[43]刘裔宏.关于广义特征值估计的一个Gerschgorin型定理[J].工科数学, 1994,10:49- 51.
[44]李仁仓.广义特征值的扰动界限[J].计算数学,1989,5(2):196-204.
[45] Jiang T S, Wei M S. Equality constrained least squares problem over quaternion field [J]. Applied Mathematics Letters, 2003, 16:883—888.
[46]屠伯埙.Schur定理在四元数体上的推广[J].数学年刊.1988,9A(2):130-138.
[47]姜同松,魏木生.四元数矩阵的实表示与四元数矩阵方程[J].数学物理学报,2006,20A(4):578-584.
[48] Rong Huang. Some inequalities for the Hadamard product and the Fan product of matrices [J]. Linear Algebra Appl. 2008, 428:1551–1558.
[49]杨忠鹏.关于四元数体上行列式的注记[J].新疆大学学报(自然科学版),1997,14(2):35-39.
[50] Johnson C.R. Positive definite matrices[J].Amer Math Monthly, 1970,77(1):259-264.
[51] Erxiong Jiang, Xingzhi Zhan. Lower bounds for the spread of a Hermitian matrix [J]. Linear Algebra Appl. 1997, 256: 153–163.
[52] Ning Long, Fengli Zhang. Novel Newton’s learning algorithm of neural networks [J]. Journal of Systems Engineering and Electronics, 2006, 17 (2): 450-454.
[53] A. Berman, R.J. Plemmons. Nonnegative Matrices in the Mathematical Sciences [M]. Academic Press, New York, 1979.
[54] D.Z.Dokovic, B.H.Smith. Quaternionic matrices: Unitary similarity, simultaneous traingula- rization and some trace identities [J]. Linear Algebra Appl. 2008, 428:890-910.
[55]屠伯埙.亚正定阵理论(I) [J].数学学报,1990,33(4):462-471.
[56]庄瓦金.四元数体上的矩阵方程[J].数学学报,1987,30(5):688-694.
[57] S.L. Adler. Quaternionic Quantum Mechanics and Quantum Fields [M]. Oxford University Press, New York, 1995.
[58]王卿文,薛有才.体与环上的矩阵方程[M].北京:知识出版社,1996.
[59] Liping Huang, Zhuojun Liu. Similarity reduction of matrix over a quaternion division ring [J]. Linear Algebra Appl. 2008, 428:765-780.
[60]伍俊良.四元数体上矩阵的弱直积与弱圈积德特征值和奇异值不等式[J].重庆师范学院学报(自然科学版),1994,11(4):60-65.
[61] P.M. Cohn. Skew Field Constructions [M]. London Mathematical Society Lecture Note Series,Cambridge University Press, Cambridge, 1977.
[62] F. Zhang. Quaternions and matrices of quaternions [J]. Linear Algebra Appl. 1997,251(1):21–57.
[63] L.P. Huang. On two questions about quaternion matrices [J]. Linear Algebra Appl. 2000 ,318(2):79–86.
[64] Q.W.Wang, The General Solution to a system of real quaternion matrix equation [J]. Computer and Mathematics with Applications .2005,49(1):665-675.
[65] Q.W. Wang. A system of four matrix equations over von Neumann regular rings and it applications [J]. Acta, Math, Sinica , English series .2005,21(4):323-334.
[66] Huang Liping.The matrix equation AXB-CXD=E over the quaternion field[J]. Linear Algebra Appl. 1996,234(2):197-208.