JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2015, Vol. 50 ›› Issue (12): 15-22.doi: 10.6040/j.issn.1671-9352.0.2014.549

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Extremal eigenvalues of a class of tridiagonal interval matrices

JIAN Yuan, LIU Ding-you   

  1. School of Mathematics and Statistics, Wuhan University, Wuhan 430072, Hubei, China
  • Received:2014-12-03 Revised:2015-03-06 Online:2015-12-20 Published:2015-12-23

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Abstract: A class of tridiagonal interval matrices is studied by the recursive characteristic polynomials and Chebyshev polynomials where some results on eigenvalues are involved. The sharp (lower and upper) bound for the eigenvalues of this class of tridiagonal interval matrices is presended, and the realization matrices whose smallest (largest) eigenvalues reach the lower(upper) bound is characterized.

Key words: recursion, eigenvalue, tridiagonal interval matrix, Chebyshev polynomial

CLC Number: 

  • O151.21
[1] KULKARNI D, SCHMIDT D, TSUI S K. Eigenvalues of tridiagonal pseudo-Toeplitz matrices[J]. Linear Algebra and its Applications, 1999, 297(1):63-80.
[2] DA FONSECA C M. On the location of the eigenvalues of Jacobi matrices[J]. Applied Mathematics Letters, 2006, 19(11):1168-1174.
[3] GOVER M J C. The eigenproblem of a tridiagonal 2-Toeplitz matrix[J]. Linear Algebra and its Applications, 1994, 197:63-78.
[4] BUCHHOLZER H, KANZOW C. Bounds for the extremal eigenvalues of a class of symmetric tridiagonal matrices with applications[J]. Linear Algebra and its Applications, 2012, 436(7):1837-1849.
[5] ZHAN Xingzhi. Extremal eigenvalues of real symmetric matrices with entries in an interval[J]. SIAM Journal on Matrix Analysis and Applications, 2005, 27(3):851-860.
[6] LENG Huinan, HE Zhiqing. Computing eigenvalue bounds of structures with uncertain but non random parameters by a method based on perturbation theory[J]. Communications in Numerical Methods in Engineering, 2007, 23(11):973-982.
[7] HORN R A, JOHNSON C R. Matrix analysis[M]. Cambridge: Cambridge University Press, 2012.
[8] YUAN Quan, LENG Huinan, HE Zhiqing. A property of eigenvalue bounds for a class of symmetric tridiagonal interval matrices[J]. Numerical Linear Algebra with Applications, 2011, 18(4):707-717.
[9] MOORE R E, KEARFOTT R B, Cloud M J. Introduction to interval analysis[M]. Cambridge: Cambridge University Press, 2009.
[10] GELFAND I. Normierte ringe[J]. Rech. Math. [Mat. Sbornik] N.S., 1941, 9(1):3-24.
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