一类矩阵特征值的扰动
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摘要
本篇文章主要研究了矩阵特征值的扰动,并给出了一些矩阵的绝对和相对扰动边界。
矩阵扰动分析主要是研究矩阵元素的变化对于矩阵问题的解的影响,它不仅仅与矩阵论和算子理论密切相关,而且对于矩阵计算同样有重要的意义。
矩阵扰动中的矩阵特征值问题不仅可直接解决数学中诸如非线性规划、优化、常微分方程,以及各类数学计算问题,而且在结构力学、工程设计、计算物理和量子力学中具有重要作用,目前矩阵特征值问题的应用大多来自于解数学物理方程、差分方程、Markov过程等。正因为它具有重要意义和广泛的应用,所以矩阵特征值扰动问题是具有深刻理论意义和广泛应用背景的研究任务之一。
矩阵特征值扰动理论在上个世纪后半叶得到充分的发展,国外的发展体系比较完善,建立了矩阵特征值扰动理论的基本框架,国内在上世纪80年代中期以后,一批致力于基础理论研究的数学工作者在这一领域取得了长足的发展,使矩阵特征值扰动理论的分析方法、研究范围都有突破性的进展,为其在其他学科上的应用起到了导向和借鉴作用。
本文研究了矩阵特征值的加法扰动问题,给出了矩阵特征值的加法扰动的新的绝对扰动边界和相对扰动边界。
文章研究问题如下：
首先本文介绍了国内外矩阵特征值的扰动分析的研究现状和矩阵特征值扰动的预备知识等。
其次,利用矩阵特征值的奇异值分解,给出了特殊矩阵,可对角化矩阵,可对称化矩阵的Wielandt-hoffman和Weyl定理的绝对扰动边界,得到了进一步的加强的结果,分析了任意矩阵的特征值的扰动,得到了任意矩阵特征值的扰动的Weyl型定理：利用矩阵的Schur分解,分析了任意矩阵的扰动,得到了新的任意矩阵的扰动边界。
最后,给出了特殊矩阵、可对角化矩阵的相对扰动界；利用矩阵的Schur三角分解,进而得到任意矩阵新的相对扰动界。
The purpose of this paper is to study the additive perturbation of matrix eigenvalue. Some new absolute and relative perturbation bounds of matrix are obtained.
Matrix perturbation analysis mainly studies the effect that the variances of matrix elements influence the sequence of matrix. It is not only relevant with theory of matrix and theory of operator, but also is important to matrix count.
The eigenvalue problem of matrix perturbation not only cope to problem of mathematic count, such as linear programming, optimization, differential equations, but also have important applications in structural mechanics, control design, computational physics and quantum mechanics. Presently, in most cases, the matrix eigenvalue is applied in solving the equation of mathematical physics, difference equation, and Markov process and so on. As it has important significance and comprehensive application, the eigenvalue problem of matrix perturbation is one of research projects which has rich theoretical sense and comprehensive application background.
The theory of matrix eigenvalue perturbation gains an adequate development in the latter half of the last century. Overseas systems are relative perfect, and establish the basic framework of the theory of matrix eigenvalue perturbation. Since the mid-eighties of the last century, a batch of domestic academician who devote to basic study, have made great strides. At analytical method and field of research of the theory of matrix eigenvalue perturbation, there are quantum jumps which would have oriented and quotable effects when be applied to other subjects.
The details will go as follows:
Firstly the current study situation and basic knowledge of matrix eigenvalue are introduced.
Secondly we improve some previous corresponding results in some sense.
Using the singular value decomposition, Some new Wielandt-Hoffiman type and Weyl type absolute perturbation bounds of special matrix, symmetrical matrix diagonalizable matrix are obtained. We also analyze the arbitrary perturbation of normal matrix, based on we give Weyl type absolute perturbation bounds of normal matrix. Using the schur triangular factorization of matrix, we extend the absolute perturbation bounds of arbitrary matrix. We use some new methods to get results, which improve and extend the corresponding results in other Papers.
Lastly, the new relative perturbation bounds for special matrix and diagonalizable matrix are obtained. Using the schur triangular factorization of matrix, we deal with the relative Perturbation bounds of arbitrary matrix and give some new theorems.
矩阵扰动分析主要是研究矩阵元素的变化对于矩阵问题的解的影响,它不仅仅与矩阵论和算子理论密切相关,而且对于矩阵计算同样有重要的意义。
矩阵扰动中的矩阵特征值问题不仅可直接解决数学中诸如非线性规划、优化、常微分方程,以及各类数学计算问题,而且在结构力学、工程设计、计算物理和量子力学中具有重要作用,目前矩阵特征值问题的应用大多来自于解数学物理方程、差分方程、Markov过程等。正因为它具有重要意义和广泛的应用,所以矩阵特征值扰动问题是具有深刻理论意义和广泛应用背景的研究任务之一。
矩阵特征值扰动理论在上个世纪后半叶得到充分的发展,国外的发展体系比较完善,建立了矩阵特征值扰动理论的基本框架,国内在上世纪80年代中期以后,一批致力于基础理论研究的数学工作者在这一领域取得了长足的发展,使矩阵特征值扰动理论的分析方法、研究范围都有突破性的进展,为其在其他学科上的应用起到了导向和借鉴作用。
本文研究了矩阵特征值的加法扰动问题,给出了矩阵特征值的加法扰动的新的绝对扰动边界和相对扰动边界。
文章研究问题如下：
首先本文介绍了国内外矩阵特征值的扰动分析的研究现状和矩阵特征值扰动的预备知识等。
其次,利用矩阵特征值的奇异值分解,给出了特殊矩阵,可对角化矩阵,可对称化矩阵的Wielandt-hoffman和Weyl定理的绝对扰动边界,得到了进一步的加强的结果,分析了任意矩阵的特征值的扰动,得到了任意矩阵特征值的扰动的Weyl型定理：利用矩阵的Schur分解,分析了任意矩阵的扰动,得到了新的任意矩阵的扰动边界。
最后,给出了特殊矩阵、可对角化矩阵的相对扰动界；利用矩阵的Schur三角分解,进而得到任意矩阵新的相对扰动界。
The purpose of this paper is to study the additive perturbation of matrix eigenvalue. Some new absolute and relative perturbation bounds of matrix are obtained.
Matrix perturbation analysis mainly studies the effect that the variances of matrix elements influence the sequence of matrix. It is not only relevant with theory of matrix and theory of operator, but also is important to matrix count.
The eigenvalue problem of matrix perturbation not only cope to problem of mathematic count, such as linear programming, optimization, differential equations, but also have important applications in structural mechanics, control design, computational physics and quantum mechanics. Presently, in most cases, the matrix eigenvalue is applied in solving the equation of mathematical physics, difference equation, and Markov process and so on. As it has important significance and comprehensive application, the eigenvalue problem of matrix perturbation is one of research projects which has rich theoretical sense and comprehensive application background.
The theory of matrix eigenvalue perturbation gains an adequate development in the latter half of the last century. Overseas systems are relative perfect, and establish the basic framework of the theory of matrix eigenvalue perturbation. Since the mid-eighties of the last century, a batch of domestic academician who devote to basic study, have made great strides. At analytical method and field of research of the theory of matrix eigenvalue perturbation, there are quantum jumps which would have oriented and quotable effects when be applied to other subjects.
The details will go as follows:
Firstly the current study situation and basic knowledge of matrix eigenvalue are introduced.
Secondly we improve some previous corresponding results in some sense.
Using the singular value decomposition, Some new Wielandt-Hoffiman type and Weyl type absolute perturbation bounds of special matrix, symmetrical matrix diagonalizable matrix are obtained. We also analyze the arbitrary perturbation of normal matrix, based on we give Weyl type absolute perturbation bounds of normal matrix. Using the schur triangular factorization of matrix, we extend the absolute perturbation bounds of arbitrary matrix. We use some new methods to get results, which improve and extend the corresponding results in other Papers.
Lastly, the new relative perturbation bounds for special matrix and diagonalizable matrix are obtained. Using the schur triangular factorization of matrix, we deal with the relative Perturbation bounds of arbitrary matrix and give some new theorems.
引文
[1]A. J. Hoffman, H. W. Wielandt. The Variation of the Spectrum of a Normal Matrix. DUKE Math.J.,1953,20:37～39
[2]G. W. Stewart. A Note on Non-Hermitian Perturlations of Hermitian Matrice.CAN-14. AD.745006,1972
[3]W. M. Kahan. Spectra of Nearly Hermitian Matriees. Proe.Amer.Math.Soe.,2002,48:11～17
[4]W. Li, Sun W., The Perturbation Bounds for Eigenvalues of Normal Matrices. Numer. Linear Algebra APPI.,2005,12:89～94
[5]吕炯兴.关于Hermite矩阵的任意扰动.南京航空航天大学学报,1998,30(2)：121～125
[6]莫华荣,黎稳.Hermite矩阵特征值的新扰动界.应用数学学报,2006,29(6)：1033～1038
[7]吕炯兴.可对称化矩阵特征值的扰动.南京航空航天大学学报,1994,26(3)：384～388
[8]吕炯兴.可对称化矩阵特征值的扰动界.高等学校计算数学学报,1994,(2)：177～185
[9]宋永忠.一类矩阵特征值的扰动.纯粹数学与应用数学,1992,8(2)：40～43
[10]张振跃.关于非亏损矩阵特征值的扰动.计算数学,1986,8(1)：106～108
[11]吕炯兴.几个矩阵范数不等式及其在谱扰动中的应用.高等学校计算数学报,2001,(2)：162～170
[12]孙继广.关于Wielandt-Hoffman定理.计算数学,1983,(2)：208～212
[13]吕炯兴.正规矩阵的任意扰动.高等学校计算数学学报,2000,(1)：85～89
[14]宋永忠.任意矩阵的特征值的扰动估计.应用数学,1992,5(4)：19～25
[15]施吉林,肖丁.任意矩阵特征值扰动的估计.高等学校计算数学学报,1987,(2)：190～192
[16]C. Eisenstat. Three Absolute Pertulbation Bounds for Matrix Eigenvalue Simply Relative Bounds. Siamj. Matrix Anal.Appi.,2008,20(1):149～158
[17]L. Elsener, S. Friendland. Singular Values, Doubly Stoehastic Matrices and Applications. Linear Algebra Appi,1995,220:161～169
[18]R. C. Li. Relative Perturbation theory (Ⅰ):Eigenvalue and Singular Value Variations. Siam J. Matrixanal. Appi.,1998,19(4):956～982
[19]R. C. Li. Relative Peturbation Theory (Ⅰ):Eigenvalue Variations. University of Tennessee. Knoxville.1997
[20]易大义,陈道琦.数值分析引论.浙江：浙江大学出版社.2008：280～285
[21]蒋正新,施国梁.矩阵理论及其应用.北京：北京航空学院出版社.1988：95～99
[22]熊洪允,曾绍标,毛云英等.应用数学基础.天津：天津大学出版社.1994：72～74
[23]孙继广.矩阵扰动分析.北京：科学出版社.2001：1～226
[24]H. Weyl. Das Asymtotische Verteilungsgesetz Der Eigenwerte Linearer Partieller Differentialgleichungen. Math. Alin.1912,71:441～479
[25]G. M. Krause. Bounds for the Variation of Matrix Eigenvalues and Polynomial Root. Inear Algebra Appl,2004,208:73～82
[26]R. Hatia, F. Kittanch, Li Ren-Cang. Some Inequalities for Commutators and Aplication to Spectral Variatioll. Linear and Multilinearalgebra,1997,43:207～220
[27]孙继广.关于正规矩阵特征值的扰动.计算数学,1954,3：334～336
[28]蒋尔雄.对称矩阵计算.上海：上海科学技术出版社.1984
[29]谈雪媛.关于方阵特征值扰动的两个注记.南京师大学报,2002,25(4)：17～19
[30]陈小山,陈艳美.矩阵特征值的相对扰动界.华南师范大学学报(自然科学版).2006,(4)：26～20
[31]陈小山.矩阵扰动若干问题研究.华南师范大学博士学位论文.广州.2001
[32]L. Elsener, S. Friendland. Singular Values, Doubly Stochastic Matrices and Applieations, Linear Algebra APPI. 1995,220:161～169
[33]陈建新,黎稳.任意矩阵的谱的Euelid距离的估计.工程数学学报,2005,22(1)：183～186
[34]王明辉,魏木生.关于任意矩阵谱的Euclid距离的估计.工程数学学报,2007,24(3)：555～558
[35]X. R. Wang, J. L. Zhang, J. B. Chen. All Admissible Linear Estimators of Regression Coefficients and Parameters in Variance Component Models. Acta. Math. Sinica, 2004,36:653～662
[36]A. Barrlund. Perturbation Bounds on the Polar Decomposition. Bit,1990,(31)
[37]J. L. Zhan, J. B. Chen. The Minimum Efficiency of LS in Gauss-Markov and Variance Component Models. Technical Reports Of Yunnan University,2002,3:426～435
[38]R. C. Li. A Perturbation Bound for the Generalized Polar Decomposition. Bit,2007, 33:304～308.
[39]R. C. Li. New Perturbation Bounds for the Unitary Polar Factor. SIAM J. Matrix Anal. Appl,1995,16(1)
[40]Mathias Roy. Perturbation Bounds for the Polar Decomposition. SIAM J. Matrix Anal Appl,1993,14:588～597
[2]G. W. Stewart. A Note on Non-Hermitian Perturlations of Hermitian Matrice.CAN-14. AD.745006,1972
[3]W. M. Kahan. Spectra of Nearly Hermitian Matriees. Proe.Amer.Math.Soe.,2002,48:11～17
[4]W. Li, Sun W., The Perturbation Bounds for Eigenvalues of Normal Matrices. Numer. Linear Algebra APPI.,2005,12:89～94
[5]吕炯兴.关于Hermite矩阵的任意扰动.南京航空航天大学学报,1998,30(2)：121～125
[6]莫华荣,黎稳.Hermite矩阵特征值的新扰动界.应用数学学报,2006,29(6)：1033～1038
[7]吕炯兴.可对称化矩阵特征值的扰动.南京航空航天大学学报,1994,26(3)：384～388
[8]吕炯兴.可对称化矩阵特征值的扰动界.高等学校计算数学学报,1994,(2)：177～185
[9]宋永忠.一类矩阵特征值的扰动.纯粹数学与应用数学,1992,8(2)：40～43
[10]张振跃.关于非亏损矩阵特征值的扰动.计算数学,1986,8(1)：106～108
[11]吕炯兴.几个矩阵范数不等式及其在谱扰动中的应用.高等学校计算数学报,2001,(2)：162～170
[12]孙继广.关于Wielandt-Hoffman定理.计算数学,1983,(2)：208～212
[13]吕炯兴.正规矩阵的任意扰动.高等学校计算数学学报,2000,(1)：85～89
[14]宋永忠.任意矩阵的特征值的扰动估计.应用数学,1992,5(4)：19～25
[15]施吉林,肖丁.任意矩阵特征值扰动的估计.高等学校计算数学学报,1987,(2)：190～192
[16]C. Eisenstat. Three Absolute Pertulbation Bounds for Matrix Eigenvalue Simply Relative Bounds. Siamj. Matrix Anal.Appi.,2008,20(1):149～158
[17]L. Elsener, S. Friendland. Singular Values, Doubly Stoehastic Matrices and Applications. Linear Algebra Appi,1995,220:161～169
[18]R. C. Li. Relative Perturbation theory (Ⅰ):Eigenvalue and Singular Value Variations. Siam J. Matrixanal. Appi.,1998,19(4):956～982
[19]R. C. Li. Relative Peturbation Theory (Ⅰ):Eigenvalue Variations. University of Tennessee. Knoxville.1997
[20]易大义,陈道琦.数值分析引论.浙江：浙江大学出版社.2008：280～285
[21]蒋正新,施国梁.矩阵理论及其应用.北京：北京航空学院出版社.1988：95～99
[22]熊洪允,曾绍标,毛云英等.应用数学基础.天津：天津大学出版社.1994：72～74
[23]孙继广.矩阵扰动分析.北京：科学出版社.2001：1～226
[24]H. Weyl. Das Asymtotische Verteilungsgesetz Der Eigenwerte Linearer Partieller Differentialgleichungen. Math. Alin.1912,71:441～479
[25]G. M. Krause. Bounds for the Variation of Matrix Eigenvalues and Polynomial Root. Inear Algebra Appl,2004,208:73～82
[26]R. Hatia, F. Kittanch, Li Ren-Cang. Some Inequalities for Commutators and Aplication to Spectral Variatioll. Linear and Multilinearalgebra,1997,43:207～220
[27]孙继广.关于正规矩阵特征值的扰动.计算数学,1954,3：334～336
[28]蒋尔雄.对称矩阵计算.上海：上海科学技术出版社.1984
[29]谈雪媛.关于方阵特征值扰动的两个注记.南京师大学报,2002,25(4)：17～19
[30]陈小山,陈艳美.矩阵特征值的相对扰动界.华南师范大学学报(自然科学版).2006,(4)：26～20
[31]陈小山.矩阵扰动若干问题研究.华南师范大学博士学位论文.广州.2001
[32]L. Elsener, S. Friendland. Singular Values, Doubly Stochastic Matrices and Applieations, Linear Algebra APPI. 1995,220:161～169
[33]陈建新,黎稳.任意矩阵的谱的Euelid距离的估计.工程数学学报,2005,22(1)：183～186
[34]王明辉,魏木生.关于任意矩阵谱的Euclid距离的估计.工程数学学报,2007,24(3)：555～558
[35]X. R. Wang, J. L. Zhang, J. B. Chen. All Admissible Linear Estimators of Regression Coefficients and Parameters in Variance Component Models. Acta. Math. Sinica, 2004,36:653～662
[36]A. Barrlund. Perturbation Bounds on the Polar Decomposition. Bit,1990,(31)
[37]J. L. Zhan, J. B. Chen. The Minimum Efficiency of LS in Gauss-Markov and Variance Component Models. Technical Reports Of Yunnan University,2002,3:426～435
[38]R. C. Li. A Perturbation Bound for the Generalized Polar Decomposition. Bit,2007, 33:304～308.
[39]R. C. Li. New Perturbation Bounds for the Unitary Polar Factor. SIAM J. Matrix Anal. Appl,1995,16(1)
[40]Mathias Roy. Perturbation Bounds for the Polar Decomposition. SIAM J. Matrix Anal Appl,1993,14:588～597