理论数学  >> Vol. 5 No. 5 (September 2015)

随机矩阵新的非1特征值包含集
New Sets to Localize All Eigenvalues Different from 1 for a Stochastic Matrix

DOI: 10.12677/PM.2015.55034, PDF, HTML, XML, 下载: 1,884  浏览: 5,911  国家自然科学基金支持

作者: 李素华*, 李耀堂:云南大学,数学与统计学院,云南 昆明

关键词: 随机矩阵S-SDD矩阵具有相同行和实矩阵非奇异特征值包含集Stochastic Matrices S-SDD Matrices Real Matrices with Same Row Sums Nonsingular Eigenvalue Inclusion Set

摘要: 本文利用S-SDD矩阵的非奇异性及修正矩阵理论,给出具有非零相同行和实矩阵非奇异的三个新的充分条件,进而得到了随机矩阵的三个新的非1特征值包含集。数值例子表明,所得结果改进了Shen et al. [Linear Algebra Appl., 447 (2014) 74-87],Cvetkovic et al. [ETNA., 18 (2004) 73-80]和Li et al. [Linear and Multilinear Algebra, http://dx.doi.org/10.1080/03081087.2014.986044]的结果。
Abstract: By using the nonsingularity of S-SDD matrices and the theory of modified matrices, three new suf-ficient conditions of the nonsingular real matrices with nonzero same row sums are given, and then three new sets to localize all eigenvalues different from 1 for a stochastic matrix are obtained. Numerical examples are given to illustrate that the proposed results are better than the results of Shen et al. [Linear Algebra Appl., 447(2014)74-87], Cvetkovic et al. [ETNA., 18(2004)73-80] and Li et al. [Linear and Multilinear Algebra, http://dx.doi.org/10.1080/03081087.2014.986044].

文章引用: 李素华, 李耀堂. 随机矩阵新的非1特征值包含集[J]. 理论数学, 2015, 5(5): 238-246. http://dx.doi.org/10.12677/PM.2015.55034

参考文献

[1] Seneta, E. (2004) Nonnegative matrices and Markov chains. Springer-Verlag, Berlin.
[2] Karlin, S. and Mcgregor, J. (1959) A characterization of birth and death processes. Proceedings of the National Academy of Sciences of the United States of America, 45, 247-266.
http://dx.doi.org/10.1073/pnas.45.3.375
[3] Yu, A. (2005) Mitrofanov Sensitivity and convergence of uniformly ergodic Markov Chains. Journal of Applied Probability, 42, 1003-1014.
http://dx.doi.org/10.1239/jap/1134587812
[4] Horn, R.A. and Johnson, C.R. (1986) Matrix analysis. Cambridge University Press, Cambridge.
[5] Kirkland, S. (2009) A cycle-based bound for subdominant eigenvalues of stochastic matrices. Linear Multilinear Algebra, 57, 247-266.
http://dx.doi.org/10.1080/03081080701669309
[6] Kirkland, S. (2009) Subdominant eigenvalues for stochastic matrices with given column sums. The Electronic Journal of Linear Algebra, 18, 784-800.
http://dx.doi.org/10.13001/1081-3810.1345
[7] Minc, H. (1988) Nonnegative matrices. Wiley Interscience, New York.
[8] Berman, A. and Plemmons, R.J. (1994) Nonnegative matrices in the mathematical sciences. SIAM, Philadelphia.
http://dx.doi.org/10.1137/1.9781611971262
[9] Cvetkovic, L., Kostic, V. and Pena, J.M. (2011) Eigenvalue lo-calization refinements for matrices related to positivity. The SIAM Journal on Matrix Analysis and Applications, 32, 771-784.
http://dx.doi.org/10.1137/100807077
[10] Varga, R.S. (2004) Gersgorin and his circles. Springer-Verlag, Berlin.
http://dx.doi.org/10.1007/978-3-642-17798-9
[11] Shen, S.-Q., Yu, J., Huang and T.-Z. (2014) Some classes of nonsingular matrices with applications to localize the real eigenvalues of real matrices. Linear Algebra and Its Applications, 447, 74-87.
http://dx.doi.org/10.1016/j.laa.2013.02.005
[12] Li, C.-Q., Liu, Q.-B. and Li, Y.-T. (2014) Geršgorin-type and Brauer-type eigenvalue localization sets of stochastic matrices. Linear and Multilinear Algebra.
[13] Cvetkovic, L., Kostic, V. and Varga, R.S. (2004) A new Gersgorin-type eigenvalue inclusion set. Electronic Transactions on Numerical Analysis, 18, 73-80.