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Keywords:
eigenvalues; lower and upper bounds; Stein equation
eigenvalues; lower and upper bounds; Stein equation
Summary:
This paper is concerned with bounds of eigenvalues of a complex matrix. Both lower and upper bounds of modulus of eigenvalues are given by the Stein equation. Furthermore, two sequences are presented which converge to the minimal and the maximal modulus of eigenvalues, respectively. We have to point out that the two sequences are not recommendable for practical use for finding the minimal and the maximal modulus of eigenvalues.
References:
[1] Bartels, R. H., Stewart, G. W.: Solution of the matrix equation $AX+XB=C$. Comm. ACM 15 (1972), 820–826. DOI 10.1145/361573.361582
[2] Chen, M. Q., Li, X. Z.: An estimation of the spectral radius of a product of block matrices. Linear Algebra Appl. 379 (2004), 267–275. MR 2039742 | Zbl 1043.15012
[3] Golub, G. H., Loan, C. F. Van: Matrix Computations Third edition. Johns Hopkins University Press, Baltimore 1996. MR 1417720
[4] Hall, C. A., Porsching, T. A.: Bounds for the maximal eigenvalue of a nonnegative irreducible matrix. Duke Math. J. 36 (1969), 159–164. MR 0240121
[5] Hu, G. D., Hu, G. D.: A relation between the weighted logarithmic norm of matrix and Lyapunov equation. BIT 40 (2000), 506–510.
[6] Hu, G. D., Liu, M. Z.: The weighted logarithmic matrix norm and bounds of the matrix exponential. Linear Algebra Appl. 390 (2004), 145–154. MR 2083412 | Zbl 1060.15024
[7] Hu, G. D., Liu, M. Z.: Properties of the weighted logarithmic matrix norms. IMA J. Math. Contr. Inform. 25 (2008), 75–84. MR 2410261 | Zbl 1144.15018
[8] Huang, T. H., Ran, R. S.: A simple estimation for the spectral radius of block H-matrics. J. Comput. Appl. Math. 177 (2005), 455–459. DOI 10.1016/j.cam.2004.09.059 | MR 2125329
[9] Lancaster, P.: The Theory of Matrices with Application. Academic Press, Orlando 1985. MR 0245579
[10] Lu, L. Z.: Perron complement and Perron root. Linear Algebra Appl. 341 (2002), 239–248. DOI 10.1016/S0024-3795(01)00378-0 | MR 1873622 | Zbl 0999.15009
[11] Lu, L. Z., Ng, M. K.: Localization of Perron roots. Linear Algebra its Appl. 392 (2004), 103–107. MR 2095910 | Zbl 1067.15004
[12] Mond, B., Pecaric, J. E.: On an inequality for spectral radius. Linear and Multilinear Algebra 20 (1996), 203–206. MR 1382076 | Zbl 0866.15011
[13] Yang, Z. M.: The block smoothing mehtod for estimationg the spectral radius of a nonnegative matrix. Appl. Math. Comput. 185 (2007), 266–271. DOI 10.1016/j.amc.2006.06.108 | MR 2298447
[14] Zhu, Q., Hu, G. D., Zeng, L.: Estimating the spectral radius of a real matrix by discrete Lyapunov equation. J. Difference Equations Appl. To appear.
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