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

Previous Articles     Next Articles

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

PDF (PC)

1419

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.
[1] WANG Su-yun, LI Yong-jun. Solvability of nonlinear second-order boundary value problems with nonlinearities which cross the resonance points [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(6): 53-56.
[2] ZHANG Yan-yan, YAN Chao. Approximation of lagrange interpolation polynomials based on the fourth Chebyshev nodes [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(8): 10-16.
[3] HAO Ping-ping, WEI Guang-sheng. Eigenvalue asymptotics of a class of the Dirac operator [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2015, 50(02): 55-59.
[4] WANG Feng. Estimation on lower bounds for the minimum eigenvalue of the Hadamard  product of inverse matrix of nonsingular M-matrix and M-matrix [J]. J4, 2013, 48(8): 30-33.
[5] JIN Guang-qing, ZUO Lian-cui*. An Upper Bound on the Sum of the k Largest Eigenvalues of  the Signless Laplacian Matrix of a Graph [J]. J4, 2013, 48(8): 1-4.
[6] SUN De-rong, XU Lan. Trees with exactly three main eigenvalues [J]. J4, 2013, 48(6): 23-28.
[7] LI Yu-ying1,2, ZHANG Zhi-ling1, LIN Hong-kang1,2, RUAN Qun-sheng1, ZHANG Shi-liang1. Dependence-reduction and application of outward-recursion data [J]. J4, 2013, 48(10): 41-46.
[8] ZHAO Na. The finite spectrum of SturmLiouville problems on time scales [J]. J4, 2013, 48(09): 96-102.
[9] 杨晓英,刘新. New lower bounds on the minimum eigenvalue of the Hadamard product of an M-matrix and its inverse [J]. J4, 2012, 47(8): 64-67.
[10] LI Yu-ying1,2, LIN Hong-kang1,2. Inward-recursion information and characteristics-identification of inward-recursion binary tree [J]. J4, 2012, 47(8): 86-91.
[11] CAO Hai-song， WU Jun-liang. On localization of eigenvalues in elliptic region [J]. J4, 2012, 47(10): 49-53.
[12] LIAO Wen-shi, WU Jun-liang. Some new upper bounds of the spread of a matrix [J]. J4, 2012, 47(10): 54-58.
[13] QIAN Xiao-yan. A subspace accelerated truncated Newton algorithm for solving the extreme eigenvalue of large-scale sparse symmetric matrix [J]. J4, 2011, 46(8): 8-12.
[14] GAO Jie. Structure of eigenvalues of multi-point boundary value problems [J]. J4, 2011, 46(8): 17-22.
[15] WU Yun1, YAO Yangxin2. Eigen-value problem of a bi-harmonic equation in R4 [J]. J4, 2011, 46(6): 45-48.
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
No Suggested Reading articles found!
Copyright 2018 © Editorial office of JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)
Supported by Beijing Magtech Co.Ltd