四元数矩阵特征值计算
|
摘要
四元数是在1843年由英国数学家W．R．哈密顿提出的。四元数的发现是数学史上的一个重大的事件。四元数在代数学，几何学，物理学，工程技术等方面有着广泛和重要的应用。特别是近10年以来，四元数在计算机科学，工程技术中的应用越来越多，更加受到人们的重视。
矩阵计算是科学与工程计算的核心，它包括三大问题：线性代数方程组问题；线性最小二乘问题和矩阵特征值问题。矩阵特征值问题是当前迅速发展的计算机科学和数值代数中一个活跃的研究课题，在自然科学和工程技术中有着广泛的重要的应用。
以实四元数作为其元素的矩阵称为实四元数矩阵(以下简称四元数矩阵)．关于四元数矩阵的研究，近几十年来，已取得很多成果[1]，[23]-[27]，[32]，[47]，[52]，[55]-[57]．一般讲，很多复矩阵的性质可以推广到四元数矩阵上来，但是四元数矩阵也具有独特的与复矩阵不同的性质。关于四元数矩阵的数值计算，工作较少，尤其是四元数矩阵奇异特征值的计算，基本上尚未开始研究，难度很大。解决四元数矩阵的特征值问题同样具有非常重要的意义。设A是一个四元数矩阵，若λ满足Ax=λx(Ax=xλ)，则λ称为A的奇异(右)特征值。四元数矩阵的奇异特征值和右特征值存在着很大的差别。到目前为止，关于四元数矩阵右特征值的研究已经得到了很多令人满意的结果。Bunse-Gerstner等将复矩阵的OR算法应用到四元数矩阵中[1]，给出了四元数矩阵的OR分解和Schur分解，从而得到该四元数矩阵的右特征值和右特征向量。在本文中，我们将实矩阵特征值的乘幂法推广到自共轭实四元数矩阵中，得到关于自共轭实四元数矩阵右特征值的乘幂法。四元数矩阵右特征值的计算可转化为它的复表示矩阵的特征值的计算问题，本文利用复表示矩阵的特殊结构给出了一种减少计算其特征值计算量的方法。
四元数矩阵计算中有一些新的问题是复矩阵计算中没有的内容。例如四元数奇异特征值的计算。黄礼平和So Wasin在[25]中讨论了2阶四元数矩阵的奇异特征值的性质，并给出了这些奇异特征值的代数表达式。在本文中，我们用C++编制的程序可以给出任意一个2阶四元数矩阵奇异特征值的数值解。本文还讨论了n阶实四元数矩阵奇异特征值的位置估计问题，给出关于实四元数矩阵奇异特征值的圆盘定理。本文还给出了计算一些特殊四元数矩阵一个或多个奇异特征值的方法。
Quaternion is initiated by W. R. Hamilton, a British mathematician, in 1843. The discovery of quaternion is an importance event in the history of math. Quaternion is applied widely in many domains, such as algebra, geometry, physics, engineering and so on. At present ten years, quaternion is applied more and more in computer science and engineering and gets more and more attention.
Matrix computation, the core of science and engineering computation, includes three aspects: solution of linear systems, least squares problems, eigenproblems. Eigenproblems are active problems of computer science and numerical algebra which develop rapidly and have great and wide application in science and engineering.
A matrix whose elements are real quaternions is called a real quaternionic matrix (quatemionic matrix for short). The study on quaternionic matrixes has gained great achievements over the past several years [1], [23]-[27], [32], [47], [52], [55]-[57]. Generally speaking, many properties of complex matrixes can be extended to quaternionic matrixes. But quaternionic matrixes also have properties which are different from complex matrixes. Few studies about numerical computation of quaternionic matrixes have been made. The study about the computation of singular eigenvalues of quaternionic matrixes almost hasn't been done because it is too difficult. It has great significance to study eigenproblems of quaternionic matrixes. Let A be a quaternionic matrix,λis called a singular eigenvalue (right eigenvalue) if Ax =λx(Ax = xλ). There are many differences between singular and right eigenvalues of quaternionic matrixes. The study on the right eigenvalues of quaternionic matrixes has gained many satisfied achievements. Bunse-Gerstner [1] gave the QR Factorzation and Schur Decomposition of quaternionic matrixes and got right eigenvalues and right eigenvectors of the matrixes by extending the QR method of complex matrixes to quaternionic matrixes. In this paper, we get the power method about the right eigenvalues of Hermite quaternionic matrixes by extending the power method of real matrixes to Hermite quaternionic matrixes. The computation of the eigenvalues of quaternionic matrixes can translate into the computation of the eigenvalues of its complex representation matrixes. In this paper, we utilize the specific structure of complex representation matrixes to get a method to reduce the quantity of computation of its eigenvalues.
There are some new problems about quaternionic matrixes which are not referred in the computation of complex matrixes such as the computation of the singular eigenvalues of quaternionic matrixes. Huang Liping and So Wasin [25] discussed the properties of the singular eigenvalues of quaternionic matrixes and gave the formula of the singular eigenvalues. In this paper, we give a programme programmed by C++ which can compute the singular eigenvalues of every 2×2 quaternionic matrix. In this paper, we discuss the problem of the position estimate of n×n quaternionic matrixes and give the Gerschgorin theorem about singular eigenvalues of quaternionic matrixes. We also give a method to compute the singular eigenvalues of some special quaternionic matrixes in this paper.
矩阵计算是科学与工程计算的核心，它包括三大问题：线性代数方程组问题；线性最小二乘问题和矩阵特征值问题。矩阵特征值问题是当前迅速发展的计算机科学和数值代数中一个活跃的研究课题，在自然科学和工程技术中有着广泛的重要的应用。
以实四元数作为其元素的矩阵称为实四元数矩阵(以下简称四元数矩阵)．关于四元数矩阵的研究，近几十年来，已取得很多成果[1]，[23]-[27]，[32]，[47]，[52]，[55]-[57]．一般讲，很多复矩阵的性质可以推广到四元数矩阵上来，但是四元数矩阵也具有独特的与复矩阵不同的性质。关于四元数矩阵的数值计算，工作较少，尤其是四元数矩阵奇异特征值的计算，基本上尚未开始研究，难度很大。解决四元数矩阵的特征值问题同样具有非常重要的意义。设A是一个四元数矩阵，若λ满足Ax=λx(Ax=xλ)，则λ称为A的奇异(右)特征值。四元数矩阵的奇异特征值和右特征值存在着很大的差别。到目前为止，关于四元数矩阵右特征值的研究已经得到了很多令人满意的结果。Bunse-Gerstner等将复矩阵的OR算法应用到四元数矩阵中[1]，给出了四元数矩阵的OR分解和Schur分解，从而得到该四元数矩阵的右特征值和右特征向量。在本文中，我们将实矩阵特征值的乘幂法推广到自共轭实四元数矩阵中，得到关于自共轭实四元数矩阵右特征值的乘幂法。四元数矩阵右特征值的计算可转化为它的复表示矩阵的特征值的计算问题，本文利用复表示矩阵的特殊结构给出了一种减少计算其特征值计算量的方法。
四元数矩阵计算中有一些新的问题是复矩阵计算中没有的内容。例如四元数奇异特征值的计算。黄礼平和So Wasin在[25]中讨论了2阶四元数矩阵的奇异特征值的性质，并给出了这些奇异特征值的代数表达式。在本文中，我们用C++编制的程序可以给出任意一个2阶四元数矩阵奇异特征值的数值解。本文还讨论了n阶实四元数矩阵奇异特征值的位置估计问题，给出关于实四元数矩阵奇异特征值的圆盘定理。本文还给出了计算一些特殊四元数矩阵一个或多个奇异特征值的方法。
Quaternion is initiated by W. R. Hamilton, a British mathematician, in 1843. The discovery of quaternion is an importance event in the history of math. Quaternion is applied widely in many domains, such as algebra, geometry, physics, engineering and so on. At present ten years, quaternion is applied more and more in computer science and engineering and gets more and more attention.
Matrix computation, the core of science and engineering computation, includes three aspects: solution of linear systems, least squares problems, eigenproblems. Eigenproblems are active problems of computer science and numerical algebra which develop rapidly and have great and wide application in science and engineering.
A matrix whose elements are real quaternions is called a real quaternionic matrix (quatemionic matrix for short). The study on quaternionic matrixes has gained great achievements over the past several years [1], [23]-[27], [32], [47], [52], [55]-[57]. Generally speaking, many properties of complex matrixes can be extended to quaternionic matrixes. But quaternionic matrixes also have properties which are different from complex matrixes. Few studies about numerical computation of quaternionic matrixes have been made. The study about the computation of singular eigenvalues of quaternionic matrixes almost hasn't been done because it is too difficult. It has great significance to study eigenproblems of quaternionic matrixes. Let A be a quaternionic matrix,λis called a singular eigenvalue (right eigenvalue) if Ax =λx(Ax = xλ). There are many differences between singular and right eigenvalues of quaternionic matrixes. The study on the right eigenvalues of quaternionic matrixes has gained many satisfied achievements. Bunse-Gerstner [1] gave the QR Factorzation and Schur Decomposition of quaternionic matrixes and got right eigenvalues and right eigenvectors of the matrixes by extending the QR method of complex matrixes to quaternionic matrixes. In this paper, we get the power method about the right eigenvalues of Hermite quaternionic matrixes by extending the power method of real matrixes to Hermite quaternionic matrixes. The computation of the eigenvalues of quaternionic matrixes can translate into the computation of the eigenvalues of its complex representation matrixes. In this paper, we utilize the specific structure of complex representation matrixes to get a method to reduce the quantity of computation of its eigenvalues.
There are some new problems about quaternionic matrixes which are not referred in the computation of complex matrixes such as the computation of the singular eigenvalues of quaternionic matrixes. Huang Liping and So Wasin [25] discussed the properties of the singular eigenvalues of quaternionic matrixes and gave the formula of the singular eigenvalues. In this paper, we give a programme programmed by C++ which can compute the singular eigenvalues of every 2×2 quaternionic matrix. In this paper, we discuss the problem of the position estimate of n×n quaternionic matrixes and give the Gerschgorin theorem about singular eigenvalues of quaternionic matrixes. We also give a method to compute the singular eigenvalues of some special quaternionic matrixes in this paper.
引文
[1] A. Bunse-Gerstner, R. Byers and V. Mehrmann. A Quaternion QR Algorithm. Numer. Math.,1989,55: 83-95.
[2] A. Bunse-Gerstner., R. Byers and V. Mehrmann. Numerically stable structure preserving methods for eigenproblems with special structure. Fakultat fur Mathematik, Universitat Bielefeld, Postf. 8640, D-4800 Bielefeld 1 (1987).
[3] A. Baker. Right eigenvalues for quaternionic matrices: A topological approach. Linear Algebra and its Applications, 1999, 286: 303-309.
[4] A. Bunse-Gerstner. Symplectic QR-like methods. Habilitationsschrift, Fakultat fur Mathematik, UniversitatBielefeld, Postf. 8640, D-4800 Bielefeld 1 (1986).
[5] A. Bunse-Gerstner, W. Gragg. Singular value decompositions of complex symmetric matrices. J. Comput.Appl. Math.,1998, 21: 41-54.
[6] B. N. Parlett. Global convergence of the basic QR algorithm on Hessenberg matrices. Math. Comput., 1968, 22: 803-817.
[7] B. N. Parlett. The Symmetric Eigenvalue Problem. Prentice-Hall, Englewood Cliffs, NJ, 1980.
[8] C. G Cao. The numerical radius of real quaternion matrix. J. Xinjiang Univ. 7, No. 2 (1990); also see MR 89k: 15031.
[9] C. G. J. Jacobi. Uber eine neue Auflosungart der bei Methode der kleinsten Quadrate vorkommenden linearen Gleichungen. Astronomische Nachrichten, 22 (1845). Translated by G. W. Stewart, University of Maryland, Tech. Report TR-92-42, April 1992.
[10] C. G. J. Jacobi. Ueber ein leichtes Verfahren, die in der Theorie der Sacularstorungen vorkommenden Gleichungen numerisch aufzulosen. J. Reine Angew. Math., 1846, 51-94.
[11] C. Lanczos. An iteration method for the solution of the eigenvalues problem of linear differential and integral operators. J. Res. Nat. Bur. Standards, 1950, 45: 255-282.
[12] Dennis I. Merion and Vladimir V. Sergeichuk. Littlewood's algorithm and quaternion matrices. Linear Algebra and its Applications ,1999, 298: 193-208.
[13] Douglas R. Farenick and Barbara A.F. Pidkowich. The spectral theorem in quaternions. Linear Algebra and its Applications, 2003, 371: 75-102.
[14] D. Watkins. Understanding the QR algorithm. SIAM Review, 1982, 24: 427-440.
[15] E.P. Wigner. Group theory and its application the quantum mechanics of atmic spectra. New York: Academic Press 1959.
[16] E.R. Davidson. The iterative calculation of a few of the lowest eigenvalues and corres-ponding eigenvectors of large real-symmetric matrices, J. Comput. Phys., 1975, 17:87-94.
[17] E.R. Davidson. Matrix eigenvector methods in quantum inechanics, in Methods in Computational Molecular Physics, G. H. E Diercksen and S. Wilson, eds., Reidel, Boston, 1983, 95-113.
[18] G. Birkhoff and S. MacLane. A Survey of Modern Algebra. 4th ed. Macmillan, 1977.
[19] Gerard L. G. Sleijpen and Henk A. Van der Vorst. A Jacobi-Davidson Iteration Method for Linear Eigenvalue Problems. SIAM Review, 2000, 42(2): 267-293.
[20] G.H. Golub and C. F. Van Loan. Matrix Computations, 3rd ed., The John Hopkins University Press, Baltimore, London, 1996.
[21] G.H. Golub, Zhenyue Zhang and Hongyuan Zha. Large sparse symmetric eigenvalue problems with homogeneous linear constraints: the Lanczos process with inner-outer iterations. Linear Algebra and its Applications, 2000, 309: 289-306.
[22] H.C. Lee. Eigenvalues of canonical forms of matrices with quaternion coefficients. Proc. R. I. A. 52, Sect. A, 1949.
[23] Huang Liping. The quatemion equation ∑ A~iXB_i=E. Acta Mathematica Sinica, New Series, 1998, 14(1): 91-98.
[24] Huang Liping. Consimilarity of quaternion matrices and complex matrices. Linear Algebra and Its Applications, 2001, 331: 21-30.
[25] Huang Liping and So Wasin. On the eigenvalues of a quaternionic matrix. Linear Algebra and Its Applications, 2001, 323: 105-116.
[26] Huang Liping. On two questions about quaternion matrices. Linear Algebra and Its Applications, 2000, 318: 79-86.
[27] Huang Liping. The matrix equation AXB-GXD=E over the quaternion. Linear Algebra and Its Applications, 1996, 234: 197-208.
[28] I. Niven. Equations in quaternions. Amer. Math. Monthly, 1995, 48: 654-661.
[29] Francis, J.. The QR transformation. Part Ⅱ. Comput. J.,1962, 5: 332-345.
[30] J.H. Van Lenthe and P. Pulay. A space-saving modification of Davidson's eigenvector algorithm. J. Comput. Chem., 1990, 11: 1164-1168.
[31] J.H. Wilkinson. The algebraic eigenvalue ptoblem. London: Oxford University Press (Clarendon) 1965.
[32] J.L. Brenner. Matrices of quaternions. Pacific J. Math., 1951, 1: 329-335.
[33] L. S. Siu. A study of polynomials, determinants, eigenvalues and numerical ranges over quaternions. M. Phil. thesis, University of Hong Kong, 1997.
[34] L. X. Chen. The extension of Cay-Hamilton theorem over the quaternion field. Chinese Sci. Bull.,1991,17:1291-1293.
[35] L. X. Chen. Definition of determinant and Cramer solutions over the quaternion field. Acta Math. Sinica N. S.,1991,7 (2): 171-180.
[36] L. X. Chen. Inverse matrix and properties of double determinant over quaternion field. Sci. China Ser. A, 1991, 34 (5): 528-540.
[37] M. Crouzeix, B. Philippe, and M. Sadkane. The Davidson method. SIAM J. Sci. Comput., 1994,15: 62-76.
[38] Nicholas J. Higham. QR factorization with complete pivoting and accurate computation of the SVD. Linear Algebra and its Applications,2000, 309 : 153-174.
[39] N. Jacobson. Basic Algebra I. W. H. Freeman, 1974.
[40] Rosch, N.. Time-reversal symmetry, Kramers' degeneracy and the algebraic eigenvalue problem. Chemical Phys.,1983, 80:1-5.
[41] N. Steenrod. The Topology of Fibre Bundles. Princeton, U. P., 1951.
[42] P. M. Cohn. Skew Field Constructions. London Mathematical Society Note Series, Cambridge University, Cambridge, 1977.
[43] P. M. Cohn. Skew-Field------Theory of general division rings. Cambridge University Press, 1995.
[44] R. A. Horn and C. R. Johnson. Matrix Analysis. Cambridge University Press, 1986.
[45] R. B. Morgan. Generalizations of Davidson's method for computing eigenvalues of large nonsymmetric matrices. J. Comput. Phys., 1992,101 : 287-291.
[46] R. B. Morgan and D. S. Scott. Preconditioning the Lanczos algorithm for sparse symmetric eigenvalue problems. SIAM J. Sci. Comput., 1993,14: 585-593.
[47] R. M. W. Wood. Quaternionic eigenvalues. Bull. London Math. Soc, 1985, 17: 137-138.
[48] S. L. Adler. Quaternionic Quantum Mechanics and Quantum Fields. Oxford U. P., New York, 1994.
[49] V. V. Prasolov. Problems and Theorems in Linear Algebra. American Mathematical Society Providence, Rhode Island, 1994.
[50] Y. H. Au-Yeung. On the convexity of numerical range in quaternionic Hilbert spaces. Linear and Multilinear Algebra, 1984, 16: 93-100.
[51] Y. Saad. Numerical Methods for Large Eigenvalue Problems. Manchester University Press, Manchester, UK, 1992.
[52] Zhang Fuzhen. Quaternions and Matrices of Quaternions. Linear Algebra and Its Applications，1997，251：21-57．
[53] 贾仲孝．解大规模矩阵特征问题的复合正交投影方法．中国科学(A辑)，1999，第29卷，第3期：224-232．
[54] 聂灵沼，丁石孙．代数学引论．高等教育出版社，2000．
[55] 黄礼平．四元数自共轭矩阵乘积的相似变换．数学进展，2003，32(4)：429-434．
[56] 黄礼平．广义四元数矩阵的最小多项式．数学学报，1995，38(5)：670-675．
[57] 黄礼平．自共轭四元数矩阵的极值定理．数学研究与评论，1997，17(1)：101-104．
[58] 庄瓦金．体上矩阵理论导引．科学出版社，2006．
[2] A. Bunse-Gerstner., R. Byers and V. Mehrmann. Numerically stable structure preserving methods for eigenproblems with special structure. Fakultat fur Mathematik, Universitat Bielefeld, Postf. 8640, D-4800 Bielefeld 1 (1987).
[3] A. Baker. Right eigenvalues for quaternionic matrices: A topological approach. Linear Algebra and its Applications, 1999, 286: 303-309.
[4] A. Bunse-Gerstner. Symplectic QR-like methods. Habilitationsschrift, Fakultat fur Mathematik, UniversitatBielefeld, Postf. 8640, D-4800 Bielefeld 1 (1986).
[5] A. Bunse-Gerstner, W. Gragg. Singular value decompositions of complex symmetric matrices. J. Comput.Appl. Math.,1998, 21: 41-54.
[6] B. N. Parlett. Global convergence of the basic QR algorithm on Hessenberg matrices. Math. Comput., 1968, 22: 803-817.
[7] B. N. Parlett. The Symmetric Eigenvalue Problem. Prentice-Hall, Englewood Cliffs, NJ, 1980.
[8] C. G Cao. The numerical radius of real quaternion matrix. J. Xinjiang Univ. 7, No. 2 (1990); also see MR 89k: 15031.
[9] C. G. J. Jacobi. Uber eine neue Auflosungart der bei Methode der kleinsten Quadrate vorkommenden linearen Gleichungen. Astronomische Nachrichten, 22 (1845). Translated by G. W. Stewart, University of Maryland, Tech. Report TR-92-42, April 1992.
[10] C. G. J. Jacobi. Ueber ein leichtes Verfahren, die in der Theorie der Sacularstorungen vorkommenden Gleichungen numerisch aufzulosen. J. Reine Angew. Math., 1846, 51-94.
[11] C. Lanczos. An iteration method for the solution of the eigenvalues problem of linear differential and integral operators. J. Res. Nat. Bur. Standards, 1950, 45: 255-282.
[12] Dennis I. Merion and Vladimir V. Sergeichuk. Littlewood's algorithm and quaternion matrices. Linear Algebra and its Applications ,1999, 298: 193-208.
[13] Douglas R. Farenick and Barbara A.F. Pidkowich. The spectral theorem in quaternions. Linear Algebra and its Applications, 2003, 371: 75-102.
[14] D. Watkins. Understanding the QR algorithm. SIAM Review, 1982, 24: 427-440.
[15] E.P. Wigner. Group theory and its application the quantum mechanics of atmic spectra. New York: Academic Press 1959.
[16] E.R. Davidson. The iterative calculation of a few of the lowest eigenvalues and corres-ponding eigenvectors of large real-symmetric matrices, J. Comput. Phys., 1975, 17:87-94.
[17] E.R. Davidson. Matrix eigenvector methods in quantum inechanics, in Methods in Computational Molecular Physics, G. H. E Diercksen and S. Wilson, eds., Reidel, Boston, 1983, 95-113.
[18] G. Birkhoff and S. MacLane. A Survey of Modern Algebra. 4th ed. Macmillan, 1977.
[19] Gerard L. G. Sleijpen and Henk A. Van der Vorst. A Jacobi-Davidson Iteration Method for Linear Eigenvalue Problems. SIAM Review, 2000, 42(2): 267-293.
[20] G.H. Golub and C. F. Van Loan. Matrix Computations, 3rd ed., The John Hopkins University Press, Baltimore, London, 1996.
[21] G.H. Golub, Zhenyue Zhang and Hongyuan Zha. Large sparse symmetric eigenvalue problems with homogeneous linear constraints: the Lanczos process with inner-outer iterations. Linear Algebra and its Applications, 2000, 309: 289-306.
[22] H.C. Lee. Eigenvalues of canonical forms of matrices with quaternion coefficients. Proc. R. I. A. 52, Sect. A, 1949.
[23] Huang Liping. The quatemion equation ∑ A~iXB_i=E. Acta Mathematica Sinica, New Series, 1998, 14(1): 91-98.
[24] Huang Liping. Consimilarity of quaternion matrices and complex matrices. Linear Algebra and Its Applications, 2001, 331: 21-30.
[25] Huang Liping and So Wasin. On the eigenvalues of a quaternionic matrix. Linear Algebra and Its Applications, 2001, 323: 105-116.
[26] Huang Liping. On two questions about quaternion matrices. Linear Algebra and Its Applications, 2000, 318: 79-86.
[27] Huang Liping. The matrix equation AXB-GXD=E over the quaternion. Linear Algebra and Its Applications, 1996, 234: 197-208.
[28] I. Niven. Equations in quaternions. Amer. Math. Monthly, 1995, 48: 654-661.
[29] Francis, J.. The QR transformation. Part Ⅱ. Comput. J.,1962, 5: 332-345.
[30] J.H. Van Lenthe and P. Pulay. A space-saving modification of Davidson's eigenvector algorithm. J. Comput. Chem., 1990, 11: 1164-1168.
[31] J.H. Wilkinson. The algebraic eigenvalue ptoblem. London: Oxford University Press (Clarendon) 1965.
[32] J.L. Brenner. Matrices of quaternions. Pacific J. Math., 1951, 1: 329-335.
[33] L. S. Siu. A study of polynomials, determinants, eigenvalues and numerical ranges over quaternions. M. Phil. thesis, University of Hong Kong, 1997.
[34] L. X. Chen. The extension of Cay-Hamilton theorem over the quaternion field. Chinese Sci. Bull.,1991,17:1291-1293.
[35] L. X. Chen. Definition of determinant and Cramer solutions over the quaternion field. Acta Math. Sinica N. S.,1991,7 (2): 171-180.
[36] L. X. Chen. Inverse matrix and properties of double determinant over quaternion field. Sci. China Ser. A, 1991, 34 (5): 528-540.
[37] M. Crouzeix, B. Philippe, and M. Sadkane. The Davidson method. SIAM J. Sci. Comput., 1994,15: 62-76.
[38] Nicholas J. Higham. QR factorization with complete pivoting and accurate computation of the SVD. Linear Algebra and its Applications,2000, 309 : 153-174.
[39] N. Jacobson. Basic Algebra I. W. H. Freeman, 1974.
[40] Rosch, N.. Time-reversal symmetry, Kramers' degeneracy and the algebraic eigenvalue problem. Chemical Phys.,1983, 80:1-5.
[41] N. Steenrod. The Topology of Fibre Bundles. Princeton, U. P., 1951.
[42] P. M. Cohn. Skew Field Constructions. London Mathematical Society Note Series, Cambridge University, Cambridge, 1977.
[43] P. M. Cohn. Skew-Field------Theory of general division rings. Cambridge University Press, 1995.
[44] R. A. Horn and C. R. Johnson. Matrix Analysis. Cambridge University Press, 1986.
[45] R. B. Morgan. Generalizations of Davidson's method for computing eigenvalues of large nonsymmetric matrices. J. Comput. Phys., 1992,101 : 287-291.
[46] R. B. Morgan and D. S. Scott. Preconditioning the Lanczos algorithm for sparse symmetric eigenvalue problems. SIAM J. Sci. Comput., 1993,14: 585-593.
[47] R. M. W. Wood. Quaternionic eigenvalues. Bull. London Math. Soc, 1985, 17: 137-138.
[48] S. L. Adler. Quaternionic Quantum Mechanics and Quantum Fields. Oxford U. P., New York, 1994.
[49] V. V. Prasolov. Problems and Theorems in Linear Algebra. American Mathematical Society Providence, Rhode Island, 1994.
[50] Y. H. Au-Yeung. On the convexity of numerical range in quaternionic Hilbert spaces. Linear and Multilinear Algebra, 1984, 16: 93-100.
[51] Y. Saad. Numerical Methods for Large Eigenvalue Problems. Manchester University Press, Manchester, UK, 1992.
[52] Zhang Fuzhen. Quaternions and Matrices of Quaternions. Linear Algebra and Its Applications，1997，251：21-57．
[53] 贾仲孝．解大规模矩阵特征问题的复合正交投影方法．中国科学(A辑)，1999，第29卷，第3期：224-232．
[54] 聂灵沼，丁石孙．代数学引论．高等教育出版社，2000．
[55] 黄礼平．四元数自共轭矩阵乘积的相似变换．数学进展，2003，32(4)：429-434．
[56] 黄礼平．广义四元数矩阵的最小多项式．数学学报，1995，38(5)：670-675．
[57] 黄礼平．自共轭四元数矩阵的极值定理．数学研究与评论，1997，17(1)：101-104．
[58] 庄瓦金．体上矩阵理论导引．科学出版社，2006．