矩阵的特征值定位和非奇异性判定
|
摘要
通过将复方阵A分裂为A=sI-B(其中:s为任意复数;I为单位矩阵;B为复方阵),利用矩阵非奇异性判定已有的方法,得到了A的含有两个参数(s和正整数k)的特征值包含集和非奇异性的判定方法,并证明所得特征值包含集和非奇异性判定方法比已有结果更精确、更具一般性.数值结果表明,通过调节s和k,可以对A的特征值进行更精确定位,从而判定A的非奇异性.
By splitting a complex square matrix Ainto A=sI-B,where s is an arbitrary complex number,Iis the identity matrix and Bis a complex square matrix,and by using an existing method of determination of non-singularity for matrices,some eigenvalue inclusion sets and some methods of determination of non-singularity for A with two parameters(s and a positive integer k)are obtained and proved to be more accurate and more general than some existing results.Numerical results show that by adjusting s and k,the eigenvalues of Acan be located more accurate,and the non-singularity of Acan be determined.
By splitting a complex square matrix Ainto A=sI-B,where s is an arbitrary complex number,Iis the identity matrix and Bis a complex square matrix,and by using an existing method of determination of non-singularity for matrices,some eigenvalue inclusion sets and some methods of determination of non-singularity for A with two parameters(s and a positive integer k)are obtained and proved to be more accurate and more general than some existing results.Numerical results show that by adjusting s and k,the eigenvalues of Acan be located more accurate,and the non-singularity of Acan be determined.
引文
[1]李朝迁.矩阵和高阶张量特征值的定位与估计[D].昆明:云南大学,2012:22-40.(LI Chaoqian.Localization and Estimating of Eigenvalues of Matrices and Higher Order Tensors[D].Kunming:Yunnan University,2012:22-40.)
[2]SANG Caili,ZHAO Jianxing.Eventually DSDD Matrices and Eigenvalue Localization[J/OL].Symmetry,2018-10-01.doi:10.3390/sym10/00448.
[3]SHAO Jiayu,SHAN Haiying,ZHANG Li.On Some Properties of the Determinants of Tensors[J].Linear Algebra Appl,2013,439(10):3057-3069.
[4]ZHAO Jianxing,LIU Qilong,LI Chaoqian,et al.Dashnic-Zusmanovich Type Matrices:A New Subclass of Nonsingular H-Matrices[J].Linear Algebra Appl,2018,552:277-287.
[5]SANG Caili,LI Chaoqian.Exclusion Sets in Eigenvalue Localization Sets for Tensors[J/OL].Linear Multilinear Algebra,2018-07-09.https://doi.org/10.1080/03081087.2018.1494121.
[6]LI Chaoqian,LI Yaotang.An Eigenvalue Localization Set for Tensors with Applications to Determine the Positive(Semi-)Defniteness of Tensors[J].Linear Multilinear Algebra,2016,64(4):587-601.
[7]KRESSNER D.Numerical Methods for General and Structured Eigenvalue Problems[M].Berlin:SpringerVerlag,2005:215-223.
[8]KENNETH H R.Discrete Mathematics and Its Applications[M].7th ed.New York:McGraw-Hill,2012:115-138.
[9]CVETKOVIC'L J,ERIC'M,PENA J M.Eventually SDD Matrices and Eigenvalue Localization[J].Appl Math Comput,2015,252:535-540.
[10]LIU Qiong,LI Zhengbo,LI Chaoqian.A Note on Eventually SDD Matrices and Eigenvalue Localization[J].Appl Math Comput,2017,311:19-21.
[11]MELMAN A.Gersgorin Disk Fragments[J].Math Magazine,2010,83(2):123-129.
[2]SANG Caili,ZHAO Jianxing.Eventually DSDD Matrices and Eigenvalue Localization[J/OL].Symmetry,2018-10-01.doi:10.3390/sym10/00448.
[3]SHAO Jiayu,SHAN Haiying,ZHANG Li.On Some Properties of the Determinants of Tensors[J].Linear Algebra Appl,2013,439(10):3057-3069.
[4]ZHAO Jianxing,LIU Qilong,LI Chaoqian,et al.Dashnic-Zusmanovich Type Matrices:A New Subclass of Nonsingular H-Matrices[J].Linear Algebra Appl,2018,552:277-287.
[5]SANG Caili,LI Chaoqian.Exclusion Sets in Eigenvalue Localization Sets for Tensors[J/OL].Linear Multilinear Algebra,2018-07-09.https://doi.org/10.1080/03081087.2018.1494121.
[6]LI Chaoqian,LI Yaotang.An Eigenvalue Localization Set for Tensors with Applications to Determine the Positive(Semi-)Defniteness of Tensors[J].Linear Multilinear Algebra,2016,64(4):587-601.
[7]KRESSNER D.Numerical Methods for General and Structured Eigenvalue Problems[M].Berlin:SpringerVerlag,2005:215-223.
[8]KENNETH H R.Discrete Mathematics and Its Applications[M].7th ed.New York:McGraw-Hill,2012:115-138.
[9]CVETKOVIC'L J,ERIC'M,PENA J M.Eventually SDD Matrices and Eigenvalue Localization[J].Appl Math Comput,2015,252:535-540.
[10]LIU Qiong,LI Zhengbo,LI Chaoqian.A Note on Eventually SDD Matrices and Eigenvalue Localization[J].Appl Math Comput,2017,311:19-21.
[11]MELMAN A.Gersgorin Disk Fragments[J].Math Magazine,2010,83(2):123-129.