关于M-矩阵Fan积最小特征值的不等式
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摘要
根据M-矩阵Fan积的性质,对两个M-矩阵Fan积最小特征值的下界做了进一步的研究.利用特征值包含域定理,给出两个M-矩阵Fan积最小特征值下界的新估计式.新估计式只依赖于两个M-矩阵的元素,计算简单易行.最后给出数值例子验证新估计式,提高了现有估计式的精度.
According to the properties of the Fan product of M-matrices,new lower bounds of the minimum eigenvalue for the Fan product of two M-matrices are given by using the characteristic value containing domain theorems.The new estimation formula only depend on the entries of M-matrices,therefore,they are easy to calculate.Finally,numerical example is given to show that the new bounds have improved precision of several existing results.
According to the properties of the Fan product of M-matrices,new lower bounds of the minimum eigenvalue for the Fan product of two M-matrices are given by using the characteristic value containing domain theorems.The new estimation formula only depend on the entries of M-matrices,therefore,they are easy to calculate.Finally,numerical example is given to show that the new bounds have improved precision of several existing results.
引文
[1]Horn R A,Johnson C R.Topics in Matrix Analysis[M].New York:Cambridge University Press,1991.
[2]Fang M Z.Bounds on eigenvalues of Hadamard product and the Fan product of matrices[J].Linear Algebra Appl.,2007,425(1):7-15.
[3]Liu Q B,Chen G L.On two inequalities for the Hadamard product and the Fan product of matrices[J].Linear Algebra Appl.,2009,431:974-984.
[4]Huang R.Some inequalities for the Hadamard product and the Fan product of matrices[J].Linear Algebra Appl.,2008,428:1551-1559.
[5]Liu Q B,Chen G L,Zhao L L,Some new bounds on the spectral radius of matrices[J].Linear Algebra Appl.,2010,432:936-948.
[6]Gao D J.Matrix inequalities for the Fan product and the hadamard product of matrices[J].Advance in Linear Algebra&Matrix Theory,2015(5):90-97.
[7]逄明贤.矩阵谱论[M].长春:吉林大学出版社,1989.
[8]Li Y T,Li Y Y,Wang R W,et al.Some new bounds on eigenvalues of the Hadamard product and the Fan product of matrices[J].Linear Algebra Appl.,2010,432:536-545.
[9]Guo Q P,Li H B,Leng J S.New inequalities on eigenvalues of the Hadamard product and the Fan product of matrices[J].J Inequal Appl.,2013:433-443.
[10]Chen F B,Ren X H,Hao B.Some new eigenvalue bounds for the Hadamard product and the Fan product of matrices[J].Journal of Mathematics,2014,34(5):895-903.
[11]王峰.矩阵的Hadamard积和Fan积的特征值界的新估计[J].广西师范大学学报(自然科学版),2015,33(3):66-70.Wang Feng.New bounds on eigenvalue of the Hadamard product and the Fan product of natrices[J].Journal of Guangxi Normal University(Natural Science Edition),2015,33(3):66-70.(in Chinese)
[12]孙德淑.非负矩阵Hadamard的谱半径上界和M-矩阵Fan积的最小特征值下界的新估计[J].西南师范大学学报(自然科学版),2016,41(2):7-11.Sun Deshu.New upper bound on the spectral radius for the Hadamard product and new lower bound on the minimum eigenvalue for the Fan product[J].Journal of Southwest China Normal University(Natural Science Edition).2016,41(2):7-11.(in Chinese)
[13]周平.M-矩阵及非负矩阵Hadamard积和Fan积的特征值界的估计[J].云南大学学报(自然科学版),2012,34(1):9-14.Zhou Ping.Estimate of bounds on eigenvalues of the Hadamard product and the Fan product of matrices and nonnegative matrices[J].Journal of Yunnan U-niversity(Natural Science Edition),2012,34(1):9-14.(in Chinese)
[14]Cheng G H.New bounds for eigenvalues of the Hadamard product and the Fan product of matrices[J].Taiwan J Math.,2014,18(1):305-312.
[15]王峰.非奇异M-矩阵的逆矩阵和M-矩阵的Hadamard积的最小特征值下界估计[J].应用数学,2013,26(2):341-345.Wang Feng.Estimate of bounds on the minimum eigenvalue of the Hadamard Product of inverse matrix of nonsingular M-matrix and M-matrix[J].Mathematica Applicata,2013,26(2):341-345.(in Chinese)
[2]Fang M Z.Bounds on eigenvalues of Hadamard product and the Fan product of matrices[J].Linear Algebra Appl.,2007,425(1):7-15.
[3]Liu Q B,Chen G L.On two inequalities for the Hadamard product and the Fan product of matrices[J].Linear Algebra Appl.,2009,431:974-984.
[4]Huang R.Some inequalities for the Hadamard product and the Fan product of matrices[J].Linear Algebra Appl.,2008,428:1551-1559.
[5]Liu Q B,Chen G L,Zhao L L,Some new bounds on the spectral radius of matrices[J].Linear Algebra Appl.,2010,432:936-948.
[6]Gao D J.Matrix inequalities for the Fan product and the hadamard product of matrices[J].Advance in Linear Algebra&Matrix Theory,2015(5):90-97.
[7]逄明贤.矩阵谱论[M].长春:吉林大学出版社,1989.
[8]Li Y T,Li Y Y,Wang R W,et al.Some new bounds on eigenvalues of the Hadamard product and the Fan product of matrices[J].Linear Algebra Appl.,2010,432:536-545.
[9]Guo Q P,Li H B,Leng J S.New inequalities on eigenvalues of the Hadamard product and the Fan product of matrices[J].J Inequal Appl.,2013:433-443.
[10]Chen F B,Ren X H,Hao B.Some new eigenvalue bounds for the Hadamard product and the Fan product of matrices[J].Journal of Mathematics,2014,34(5):895-903.
[11]王峰.矩阵的Hadamard积和Fan积的特征值界的新估计[J].广西师范大学学报(自然科学版),2015,33(3):66-70.Wang Feng.New bounds on eigenvalue of the Hadamard product and the Fan product of natrices[J].Journal of Guangxi Normal University(Natural Science Edition),2015,33(3):66-70.(in Chinese)
[12]孙德淑.非负矩阵Hadamard的谱半径上界和M-矩阵Fan积的最小特征值下界的新估计[J].西南师范大学学报(自然科学版),2016,41(2):7-11.Sun Deshu.New upper bound on the spectral radius for the Hadamard product and new lower bound on the minimum eigenvalue for the Fan product[J].Journal of Southwest China Normal University(Natural Science Edition).2016,41(2):7-11.(in Chinese)
[13]周平.M-矩阵及非负矩阵Hadamard积和Fan积的特征值界的估计[J].云南大学学报(自然科学版),2012,34(1):9-14.Zhou Ping.Estimate of bounds on eigenvalues of the Hadamard product and the Fan product of matrices and nonnegative matrices[J].Journal of Yunnan U-niversity(Natural Science Edition),2012,34(1):9-14.(in Chinese)
[14]Cheng G H.New bounds for eigenvalues of the Hadamard product and the Fan product of matrices[J].Taiwan J Math.,2014,18(1):305-312.
[15]王峰.非奇异M-矩阵的逆矩阵和M-矩阵的Hadamard积的最小特征值下界估计[J].应用数学,2013,26(2):341-345.Wang Feng.Estimate of bounds on the minimum eigenvalue of the Hadamard Product of inverse matrix of nonsingular M-matrix and M-matrix[J].Mathematica Applicata,2013,26(2):341-345.(in Chinese)