矩阵特征值扰动的若干问题
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摘要
矩阵的特征值问题在工程计算,物理,天文和微分方程数值求解等大规模数值计算方面起着极其重要的作用。对于给定的矩阵,要用计算机求出它的特征值,但是由于浮点误差和测量误差的存在,特征值的数值解与准确值之间必存在偏离,因此需要估计计算解的可靠性,这是矩阵扰动分析所要研究的重要课题之一。本文研究了不变子空间上矩阵特征值的绝对加法扰动和相对加法扰动,得出了几个新的扰动界。
论文主要系统地研究了特征值传统误差界的三种类型,即:Weyl型,Hoffman-Wielandt型和Bauer-Fike型,并进一步研究了几种特殊的情况,得到了加强的结果。
论文第三章研究了特征值的Weyl型扰动。在已有的原矩阵A是Hermite矩阵它的扰动矩阵B为一般矩阵得到的扰动界,以及原矩阵A和它的扰动矩阵B均为Hermite矩阵得到的扰动界的基础上,研究了Hermite在半正定条件下Weyl型的相对扰动界。
论文第四章研究了特征值的Hoffman-Wielandt型扰动。在已有的几类常见Hoffman-Wielandt扰动界和可对角矩阵特征值的Hoffman-Wielandt型扰动界的基础上,进一步研究了任意矩阵特征值的Hoffman-Wielandt型扰动界,改进了已有的一些扰动界,得到了更易计算的形式。
论文第五章研究了特征值的Bauer-Fike型扰动。在一个已有绝对扰动界的基础上,讨论了它的相对扰动界。
论文第六章研究了AQ-AB=R型扰动。通过利用矩阵的分块以及矩阵的扩充等方法,改进了原有的一些扰动界,得到了更方便的结果,并把新扰动界与原有的扰动界进行了比较。
In some large-scale numerical calculation problems , such as engineering calculation , physics,astronomy and numerical solution to differential equations, matrix eigenvalue problem plays an extremely important role. For a given matrix, if you want to use the computer to derive its eigenvalues, however, because of the existence of floating-point error and measurement error, it must exist deviation between the exact value and the numerical solution of eigenvalue, therefore , estimating the reliability of the calculation is one of the important issues to do some research to the matrix perturbation analysis .
In this paper, the absolute additive perturbation and relative additive perturbation of matrix eigenvalue in invariant subspaces are studied, and some new perturbation bounds are given. The traditional three types of error bounds: Weyl, Hoffman-Wielandt, and Bauer-Fike, are systematically studied in this article and doing some further study to several special cases, then strengthened the corresponding results.
In chapter three, some research about the Weyl type are studied. Considering the existing results of the original matrix A is the Hermite matrix and its perturbation matrix B is normal matrix , or the original matrix A and its perturbation matrix B both are Hermite matrix , some study to the relative perturbation bounds of the Weyl type , under the conditions of a positive semi-definite Hermite matrix.,which has the basis of some perturbation bounds under the conditions of a positive definite Hermite matrix.
In chapter four, some research of the Hermite-Wielandt type are studied. Considering the exsisting results of the normal matrix and the diagonal matrix , some research about the normal Hoffman-Wielandt type of normal matrix are done, and some exsit perturbation bounds are improved , some perturbation bounds are easy .
In chapter five, some research of the Bauer-Fike type are studied. A relative perturbation bound are given , which has the basis of its absolute perturbation bound.
In chapter six, some research of the AQ-QB=R type are studied. Through the use of sub-block matrix, as well as the expansion of matrix. a easy result are given ,and at the same time improving the exsit results,. In the end , some comparison are given between the new and the old results.
论文主要系统地研究了特征值传统误差界的三种类型,即:Weyl型,Hoffman-Wielandt型和Bauer-Fike型,并进一步研究了几种特殊的情况,得到了加强的结果。
论文第三章研究了特征值的Weyl型扰动。在已有的原矩阵A是Hermite矩阵它的扰动矩阵B为一般矩阵得到的扰动界,以及原矩阵A和它的扰动矩阵B均为Hermite矩阵得到的扰动界的基础上,研究了Hermite在半正定条件下Weyl型的相对扰动界。
论文第四章研究了特征值的Hoffman-Wielandt型扰动。在已有的几类常见Hoffman-Wielandt扰动界和可对角矩阵特征值的Hoffman-Wielandt型扰动界的基础上,进一步研究了任意矩阵特征值的Hoffman-Wielandt型扰动界,改进了已有的一些扰动界,得到了更易计算的形式。
论文第五章研究了特征值的Bauer-Fike型扰动。在一个已有绝对扰动界的基础上,讨论了它的相对扰动界。
论文第六章研究了AQ-AB=R型扰动。通过利用矩阵的分块以及矩阵的扩充等方法,改进了原有的一些扰动界,得到了更方便的结果,并把新扰动界与原有的扰动界进行了比较。
In some large-scale numerical calculation problems , such as engineering calculation , physics,astronomy and numerical solution to differential equations, matrix eigenvalue problem plays an extremely important role. For a given matrix, if you want to use the computer to derive its eigenvalues, however, because of the existence of floating-point error and measurement error, it must exist deviation between the exact value and the numerical solution of eigenvalue, therefore , estimating the reliability of the calculation is one of the important issues to do some research to the matrix perturbation analysis .
In this paper, the absolute additive perturbation and relative additive perturbation of matrix eigenvalue in invariant subspaces are studied, and some new perturbation bounds are given. The traditional three types of error bounds: Weyl, Hoffman-Wielandt, and Bauer-Fike, are systematically studied in this article and doing some further study to several special cases, then strengthened the corresponding results.
In chapter three, some research about the Weyl type are studied. Considering the existing results of the original matrix A is the Hermite matrix and its perturbation matrix B is normal matrix , or the original matrix A and its perturbation matrix B both are Hermite matrix , some study to the relative perturbation bounds of the Weyl type , under the conditions of a positive semi-definite Hermite matrix.,which has the basis of some perturbation bounds under the conditions of a positive definite Hermite matrix.
In chapter four, some research of the Hermite-Wielandt type are studied. Considering the exsisting results of the normal matrix and the diagonal matrix , some research about the normal Hoffman-Wielandt type of normal matrix are done, and some exsit perturbation bounds are improved , some perturbation bounds are easy .
In chapter five, some research of the Bauer-Fike type are studied. A relative perturbation bound are given , which has the basis of its absolute perturbation bound.
In chapter six, some research of the AQ-QB=R type are studied. Through the use of sub-block matrix, as well as the expansion of matrix. a easy result are given ,and at the same time improving the exsit results,. In the end , some comparison are given between the new and the old results.
引文
[1]曹志诰.矩阵特征值问题[M].高等教育出版社, 1980
[2]陈建新.可对称化矩阵特征值的任意扰动[J].电子科技大学学报, 2005, 34(1): 121-123.
[3]陈小山,陈艳美.矩阵特征值的相对扰动界[J].华南师范大学学报, 2006, 4(11): 16-20.
[4]陈小山,黎稳.正定Hermite矩阵特征值的相对扰动界[J].工程数学学报, 2003, 4(20): 140-142.
[5]陈小山,黎稳.关于特征值的Hoffman-Wielandt型相对扰动界[J].应用数学学报, 2003, 3(26): 396-401.
[6]陈小山.特征空间和奇异空间相对扰动界[J].华南师范大学学报, 2005(1): 6-10.
[7]陈小山,黎稳.极因子在酉不变范数下的相对扰动界[J].数学进展, 2006, 35(2):12-16.
[8]贾丽杰,杨虎.关于矩阵特征值的扰动[J].重庆大学学报, 2005, 28(4):113-115.
[9]黎罗罗. Hermite矩阵乘积的特征值不等式[J].中山大学学报, 1993, 32(3):14-18.
[10]刘新国.矩阵特征值的扰动[J].高等学校计算数学学报, 1987, 25(4):22-26.
[11]吕烔兴.可对称化矩阵特征值的扰动[J].南京航空航天大学学报, 1994, 26(3):384-388.
[12]吕烔兴.矩阵特征值的几个扰动定理[J].高等学校计算数学学报, 1996, 18(1):89-94.
[13]吕烔兴.关于Hermite矩阵的任意扰动[J].南京航空航天大学学报, 1998, 30(2): 121-125.
[14]吕烔兴.正规矩阵的任意扰动[J].高等学校计算数学学报, 2000, 3(1): 85-89.
[15]吕烔兴.几个矩阵范数不等式及其在谱扰动中的应用[J].高等学校计算数学学报, 2001, 6(2): 162-170.
[16]莫荣华,黎稳. Hermite矩阵特征值的新扰动界[J].应用数学学报, 2006, 29(6).
[17]宋永忠.任意矩阵的特征值的扰动估计[J].应用数学, 1992(4): 19-25.
[18]孙继广.关于Wielandt-Hoffman定理[J].计算数学, 1983, 5(2): 208-212.
[19]孙继广.关于正规矩阵的特征值的扰动[J].计算数学, 1984(6): 334-336.
[20]孙继广,陈春晖.广义极分解[J].计算数学, 1989(3).
[21]孙继广.矩阵扰动分析[M].科学出版社, 2001.
[22]谈雪媛.关于方阵特征值扰动的两个注记[J].南京师大学报, 2002, 25(4): 15-19.
[23]王伯英等. GRASSMANN空间不可合元素的若干判断方法[J].北京师范大学学报, 1987(3): 1-4.
[24]徐树方等.数值线性代数[M].北京科学出版社, 2003.
[25]姚奕,曹树喜.关于一般方阵的特征值扰动估计的一点注记[J].南京师大学报, 2003(2):20-23.
[26]张振跃.关于非亏损矩阵特征值的扰动[J].计算数学, 1986, 8(1): 106-108.
[27] A. Ben-Israel and T. Greville, Generalized Inverses: Theory and Application[J]. John Wiley ,New York, 1974
[28] A. Ben-Israel, T. N, Grevikke, Generalized Inverses[J]. Theory and Applications ,second ed., Springer-Verlag . New York .2003
[29] A,J, Hoffman , H .W. Wielandt ,The variation of the spectrum of a normal matrix[J]. Duke Math .J.20(1953) 37-39
[30] Bhatia, R, Davis, C, and M c Intosh, A, Perturbation of spectrak subspace and solution of linear operator equations[J]. Linear A Igebra Appl, 52-53,1983,45-67
[31] B.N.Parlett. The Symmetric Eigenvalue Problem Prentice-Hall. Inc. Englewood[J]. Cliffs. N.J. 1980.
[32] C.Eisenstat and Ilse C.F. Ipsen, Three Absolute Perturbation Bounds for Matrix Eigenvalues Imply Relative Bounds[J]. SLAM J. Matrix Anal .Appl ,20,1998,pp,149-158
[33] C.-H. Chen and J.-G .Sun. Perturbation bounds for the polar factors[J]. J, Comp. Math .7 ,1989,pp,397-401
[34] Daves. C .and Kahan. W. The Rotation of Eigenvectors by a Perturbation[J]. Siam J, Numer. Anak .1,1970,pp,1-46
[35] Hhation R, Kittanen F, Li .R.C .et al .Eigenvalues of symmetrizable matrices[J]. BIT ,1998, (38):1-11
[36] Eisenstat C.Ilse C F Ipsen. Three Absolute Perturbation Bounds for Matrix Eigenvalues Imply Relative Bounds[J]. SIAM J. Matrix Anal. Appl.,1998, 20(1): 149-158
[37] Elsener L, Friedland S. Singular Values, Doubly Stochastic Matrices and Applications. Linear Algebra[J]. Appl.,1995,220:161-169
[38] F. Bauer, C, Fike, Norms and exclusion theorems[J]. Numer .Math .2 (1960) 137-141
[39] G .M.. Krause, Private communications[J]. Aug ,1994
[40] Henrici, P, Bounds for Iteretes, Inverses, Spectral Variation and Fields of Value of Values of Non-normak Matrices[J]. Numer Math 1962, 24-40
[41] Hoffman A J. Weilandt H W. The Variation of the Spectrume of a Normal Matrix. Duke Math[J]. 1953, 20:37-39
[42] Horn R, Johnson C. Matrix analysis[M]. Cambridge University Press, 1985.
[43] Horn R A, Johnson C R. Topics in Matrix Analysis[J]. Cambridge: Cambridge University Press, 1991.
[44] Householder, A, S, The theory of matrices in numericak analysis[J]. Blaisedell , New York, 1964, 65-69
[45] Ilse C F Ipsen. Relative Perturbation Results for Matrix Eigenvalues and Singlular Values[J]. Acta Numerica, 1998: 151-201.
[46] J.-G. Sun, On the perturbation of the eigenvalues of a normal matrix[J]. Math. Numer .Math . 1984, 65:334-336.
[47] J.-G. Sun, Backward perturbation analysis of certain characteristic subspaces[J]. Numer. Math. 1993, 65:357-382.
[48] J.-G. Sun, On the variation of the spectrum of a normal matrix[J]. Linear Algebra and its Applications, 1996.
[49] J.H. Wilkinson. The Algebraic Eigenvalue Problem[M]. Oxford Oxford university press.Lendon, 1965.
[50] Kahan W M. Spextra of Nearly Hermitian Matrices[J]. Proc. Amer. Math. Soc. 1975,48: 11-17
[51] Li Chi-Kwong, Roy Mathias. The lidskii-mirsky-wielandt theorem additive and multiplicative versions[J]. Numer Math, 1998, 81: 377-413.
[52] Li R C. Norms of certain matrices with applications to variations of the spectra of matrices and matrix pencils[J]. Linear Algebra Appl, 1993, 182: 199-234.
[53] Li R C. A perturbation bound for the generalized polar decomposition[J]. BIT Numerical Math, 1993, 33: 304-308
[54] Li R C. Relative Perturbation Theory (I): Eigenvalue Variations[J]. LAPACK working note 84, Computer Science Department, University of Tennessee, Knoxville, Revised May, 1997.
[55] Li R C. Relative Perturbation Theory (I): Eigenvalue and Singular Value Variations[J]. SIAM J. Matrix Anal. Appl, 1998, 19:956-982.
[56] Li R C. Relative perturbation theory: (ⅱ) Eigenspace and singular subspace variations[J]. SIAM J Matrix Anal appl, 1998, 20:471-492.
[57] Li W. Sun W. The Perturbation Bounds for Eigenvalues of Normal Matrices[J]. Number Linear Algebra Appl, 2005, 12: 89-94.
[58] L. Laszlo, An attatinable lower bounds for the best normal approximation. SIAM. J. Matrix Anak[J].Appl. 15: 1035-1043, 1994.
[59] Parlett B B. The symmetric Eigenevalue Problem[M]北京科学出版社, 1980.
[60] Parlett, B.N.The Symmetric Eigenvalues Problem, Pretice-Hall, Englewood Cliffs[J]. N .J, 1980, 218-220.
[61] R.Bhatia, Perturbation Bounds for Matrix Eigenvalues[J]. Pitman Res, Notes Math .Ser. Longman Sci. Tech, Harlow, Essex, Wiley ,New York, 1987.
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[64] S.C.Eisenstat. A perturbation bound for the eigenvalues of a singular diagonalizable Matrix[J]. Linear Algebra and its Applications. 2006.
[65] Steward G W, Sun, J G. Matrix Perturbation Theory[M]. Boston : Academic Press ,1990.
[66] Wen Li and Weiwei Sun. The perturbation bounds for eigenvalues of normal matrices[J]. Linear Algebra and its Applications. 2005.
[67] Wilkinson J H.代数特征值问题[M].北京科学出版社, 2001. 96-102.
[68] W .M Kahan. Spectra of nearly Hermit matrices[J]. Proc. Amer .Math. Soc. 48: 11-17, 1975.
[69] Yongzhong Song. A note on the variation of the spectrum of an arbitrary matrix[J]. Linear Algebra and its Applications, 2002.
[70] (美)戈卢布, G.H., (美)范洛恩, C.F.著袁亚湘等译.矩阵计算[M].科学出版社, 2001.
[71] Y.Wei, X.Li, F.Bu and F.Zhang. Relative perturbation bounds for the eigenvalues of singular matrice[J]. Preprint August 2005 available from: http://www.math.umn.edu/ ~buxx001/R-perturbation.pdf.
[72] Zhang Z. On the Perturbation of Eigenvalues of a Non-defective Matrix. Math[J]. Numer. SInica, 1986, 6(1): 106-108.
[2]陈建新.可对称化矩阵特征值的任意扰动[J].电子科技大学学报, 2005, 34(1): 121-123.
[3]陈小山,陈艳美.矩阵特征值的相对扰动界[J].华南师范大学学报, 2006, 4(11): 16-20.
[4]陈小山,黎稳.正定Hermite矩阵特征值的相对扰动界[J].工程数学学报, 2003, 4(20): 140-142.
[5]陈小山,黎稳.关于特征值的Hoffman-Wielandt型相对扰动界[J].应用数学学报, 2003, 3(26): 396-401.
[6]陈小山.特征空间和奇异空间相对扰动界[J].华南师范大学学报, 2005(1): 6-10.
[7]陈小山,黎稳.极因子在酉不变范数下的相对扰动界[J].数学进展, 2006, 35(2):12-16.
[8]贾丽杰,杨虎.关于矩阵特征值的扰动[J].重庆大学学报, 2005, 28(4):113-115.
[9]黎罗罗. Hermite矩阵乘积的特征值不等式[J].中山大学学报, 1993, 32(3):14-18.
[10]刘新国.矩阵特征值的扰动[J].高等学校计算数学学报, 1987, 25(4):22-26.
[11]吕烔兴.可对称化矩阵特征值的扰动[J].南京航空航天大学学报, 1994, 26(3):384-388.
[12]吕烔兴.矩阵特征值的几个扰动定理[J].高等学校计算数学学报, 1996, 18(1):89-94.
[13]吕烔兴.关于Hermite矩阵的任意扰动[J].南京航空航天大学学报, 1998, 30(2): 121-125.
[14]吕烔兴.正规矩阵的任意扰动[J].高等学校计算数学学报, 2000, 3(1): 85-89.
[15]吕烔兴.几个矩阵范数不等式及其在谱扰动中的应用[J].高等学校计算数学学报, 2001, 6(2): 162-170.
[16]莫荣华,黎稳. Hermite矩阵特征值的新扰动界[J].应用数学学报, 2006, 29(6).
[17]宋永忠.任意矩阵的特征值的扰动估计[J].应用数学, 1992(4): 19-25.
[18]孙继广.关于Wielandt-Hoffman定理[J].计算数学, 1983, 5(2): 208-212.
[19]孙继广.关于正规矩阵的特征值的扰动[J].计算数学, 1984(6): 334-336.
[20]孙继广,陈春晖.广义极分解[J].计算数学, 1989(3).
[21]孙继广.矩阵扰动分析[M].科学出版社, 2001.
[22]谈雪媛.关于方阵特征值扰动的两个注记[J].南京师大学报, 2002, 25(4): 15-19.
[23]王伯英等. GRASSMANN空间不可合元素的若干判断方法[J].北京师范大学学报, 1987(3): 1-4.
[24]徐树方等.数值线性代数[M].北京科学出版社, 2003.
[25]姚奕,曹树喜.关于一般方阵的特征值扰动估计的一点注记[J].南京师大学报, 2003(2):20-23.
[26]张振跃.关于非亏损矩阵特征值的扰动[J].计算数学, 1986, 8(1): 106-108.
[27] A. Ben-Israel and T. Greville, Generalized Inverses: Theory and Application[J]. John Wiley ,New York, 1974
[28] A. Ben-Israel, T. N, Grevikke, Generalized Inverses[J]. Theory and Applications ,second ed., Springer-Verlag . New York .2003
[29] A,J, Hoffman , H .W. Wielandt ,The variation of the spectrum of a normal matrix[J]. Duke Math .J.20(1953) 37-39
[30] Bhatia, R, Davis, C, and M c Intosh, A, Perturbation of spectrak subspace and solution of linear operator equations[J]. Linear A Igebra Appl, 52-53,1983,45-67
[31] B.N.Parlett. The Symmetric Eigenvalue Problem Prentice-Hall. Inc. Englewood[J]. Cliffs. N.J. 1980.
[32] C.Eisenstat and Ilse C.F. Ipsen, Three Absolute Perturbation Bounds for Matrix Eigenvalues Imply Relative Bounds[J]. SLAM J. Matrix Anal .Appl ,20,1998,pp,149-158
[33] C.-H. Chen and J.-G .Sun. Perturbation bounds for the polar factors[J]. J, Comp. Math .7 ,1989,pp,397-401
[34] Daves. C .and Kahan. W. The Rotation of Eigenvectors by a Perturbation[J]. Siam J, Numer. Anak .1,1970,pp,1-46
[35] Hhation R, Kittanen F, Li .R.C .et al .Eigenvalues of symmetrizable matrices[J]. BIT ,1998, (38):1-11
[36] Eisenstat C.Ilse C F Ipsen. Three Absolute Perturbation Bounds for Matrix Eigenvalues Imply Relative Bounds[J]. SIAM J. Matrix Anal. Appl.,1998, 20(1): 149-158
[37] Elsener L, Friedland S. Singular Values, Doubly Stochastic Matrices and Applications. Linear Algebra[J]. Appl.,1995,220:161-169
[38] F. Bauer, C, Fike, Norms and exclusion theorems[J]. Numer .Math .2 (1960) 137-141
[39] G .M.. Krause, Private communications[J]. Aug ,1994
[40] Henrici, P, Bounds for Iteretes, Inverses, Spectral Variation and Fields of Value of Values of Non-normak Matrices[J]. Numer Math 1962, 24-40
[41] Hoffman A J. Weilandt H W. The Variation of the Spectrume of a Normal Matrix. Duke Math[J]. 1953, 20:37-39
[42] Horn R, Johnson C. Matrix analysis[M]. Cambridge University Press, 1985.
[43] Horn R A, Johnson C R. Topics in Matrix Analysis[J]. Cambridge: Cambridge University Press, 1991.
[44] Householder, A, S, The theory of matrices in numericak analysis[J]. Blaisedell , New York, 1964, 65-69
[45] Ilse C F Ipsen. Relative Perturbation Results for Matrix Eigenvalues and Singlular Values[J]. Acta Numerica, 1998: 151-201.
[46] J.-G. Sun, On the perturbation of the eigenvalues of a normal matrix[J]. Math. Numer .Math . 1984, 65:334-336.
[47] J.-G. Sun, Backward perturbation analysis of certain characteristic subspaces[J]. Numer. Math. 1993, 65:357-382.
[48] J.-G. Sun, On the variation of the spectrum of a normal matrix[J]. Linear Algebra and its Applications, 1996.
[49] J.H. Wilkinson. The Algebraic Eigenvalue Problem[M]. Oxford Oxford university press.Lendon, 1965.
[50] Kahan W M. Spextra of Nearly Hermitian Matrices[J]. Proc. Amer. Math. Soc. 1975,48: 11-17
[51] Li Chi-Kwong, Roy Mathias. The lidskii-mirsky-wielandt theorem additive and multiplicative versions[J]. Numer Math, 1998, 81: 377-413.
[52] Li R C. Norms of certain matrices with applications to variations of the spectra of matrices and matrix pencils[J]. Linear Algebra Appl, 1993, 182: 199-234.
[53] Li R C. A perturbation bound for the generalized polar decomposition[J]. BIT Numerical Math, 1993, 33: 304-308
[54] Li R C. Relative Perturbation Theory (I): Eigenvalue Variations[J]. LAPACK working note 84, Computer Science Department, University of Tennessee, Knoxville, Revised May, 1997.
[55] Li R C. Relative Perturbation Theory (I): Eigenvalue and Singular Value Variations[J]. SIAM J. Matrix Anal. Appl, 1998, 19:956-982.
[56] Li R C. Relative perturbation theory: (ⅱ) Eigenspace and singular subspace variations[J]. SIAM J Matrix Anal appl, 1998, 20:471-492.
[57] Li W. Sun W. The Perturbation Bounds for Eigenvalues of Normal Matrices[J]. Number Linear Algebra Appl, 2005, 12: 89-94.
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