M-矩阵最小特征值的新下界
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摘要
目的研究M-矩阵最小特征值的估计问题。方法利用Brauer定理和Gerschgorin定理,并结合不等式放缩技巧,估计M-矩阵的逆矩阵和非负矩阵的Hadamard积的谱半径上界。结果给出M-矩阵最小特征值的新下界。结论数值算例表明新估计式在一定条件下优于现有的估计式。
Purposes—To estimate the bound of the minimum eigenvalues of M-matrices.Methods—The Brauer theorem,Gerschgorin theorem and the skills of magnifying and shrinking of inequality are applied to estimate the upper bound of spectral radius of Hadamard product of inverse matrix and nonnegative matrix of M-matrices.Result—Some new lower bounds for the minimum eigenvalue of M-matrices are presented.Conclusion—Numerical examples show that these estimating formulas improve the related results in some cases.
Purposes—To estimate the bound of the minimum eigenvalues of M-matrices.Methods—The Brauer theorem,Gerschgorin theorem and the skills of magnifying and shrinking of inequality are applied to estimate the upper bound of spectral radius of Hadamard product of inverse matrix and nonnegative matrix of M-matrices.Result—Some new lower bounds for the minimum eigenvalue of M-matrices are presented.Conclusion—Numerical examples show that these estimating formulas improve the related results in some cases.
引文
[1]KRESSNER D.Numerical Methods for General and Structured Eigenvalue Problems[M].Berlin:Springer-Verlag,2005.
[2]GREEN M,LIMEBEER D J.Linear Robust Control[M].Prentice-Hall:Englewood Cliffs,NJ,2012.
[3]SHIVAKUMAR P N,WILLIAMS J J,YE Qiang,et al.On two-sided bounds related to weakly diagonally dominant M-matrices with application to digital circuit dynamics[J].SIAM Journal on Matrix Analysis and Applications,1996,17(2):298-312.
[4]TIAN Gui-xian,HUANG Ting-zhu.Inequalities for the minimum eigenvalue of M-matrices[J].Electronic Journal of Linear Algebra,2010,20:291-302.
[5]LI Chao-qian,LI Yao-tang,ZHAO Rui-juan.New inequalities for the minimum eigenvalue of M-matrices[J].Linear and Multilinear Algebra,2013,61(9):1267-1279.
[6]XU Ming,LI Su-hua,LI Chao-qian.Inequalities for the minimum eigenvalue of doubly strictly diagonally dominant M-matrices[J].Journal of Applied Mathematics,2014:535716.DOI:https://doi.org/10.1155/2014/535716.
[7]WANG Feng,SUN De-shu.Some new inequalities for the minimum eigenvalue of M-matrices[J].Journal of Inequalities and Applications,2015,195:1-7.DOI:https://doi.org/10.1186/s13660-015-0723-3.
[8]LI Yao-tang,WANG Feng,LI Chao-qian,et al.Some new bounds for the minimum eigenvalue of the Hadamard product of an M-matrix and an inverse M-matrix[J].Journal of Inequalities and Applications,2013,480:1-8.
[9]孙德淑.M-矩阵最小特征值的下界新估计[J].重庆师范大学学报(自然科学版),2016,33(2):85-89.
[2]GREEN M,LIMEBEER D J.Linear Robust Control[M].Prentice-Hall:Englewood Cliffs,NJ,2012.
[3]SHIVAKUMAR P N,WILLIAMS J J,YE Qiang,et al.On two-sided bounds related to weakly diagonally dominant M-matrices with application to digital circuit dynamics[J].SIAM Journal on Matrix Analysis and Applications,1996,17(2):298-312.
[4]TIAN Gui-xian,HUANG Ting-zhu.Inequalities for the minimum eigenvalue of M-matrices[J].Electronic Journal of Linear Algebra,2010,20:291-302.
[5]LI Chao-qian,LI Yao-tang,ZHAO Rui-juan.New inequalities for the minimum eigenvalue of M-matrices[J].Linear and Multilinear Algebra,2013,61(9):1267-1279.
[6]XU Ming,LI Su-hua,LI Chao-qian.Inequalities for the minimum eigenvalue of doubly strictly diagonally dominant M-matrices[J].Journal of Applied Mathematics,2014:535716.DOI:https://doi.org/10.1155/2014/535716.
[7]WANG Feng,SUN De-shu.Some new inequalities for the minimum eigenvalue of M-matrices[J].Journal of Inequalities and Applications,2015,195:1-7.DOI:https://doi.org/10.1186/s13660-015-0723-3.
[8]LI Yao-tang,WANG Feng,LI Chao-qian,et al.Some new bounds for the minimum eigenvalue of the Hadamard product of an M-matrix and an inverse M-matrix[J].Journal of Inequalities and Applications,2013,480:1-8.
[9]孙德淑.M-矩阵最小特征值的下界新估计[J].重庆师范大学学报(自然科学版),2016,33(2):85-89.