JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2018, Vol. 53 ›› Issue (8): 66-76.doi: 10.6040/j.issn.1671-9352.0.2018.109

Previous Articles     Next Articles

Inverse spectral problem of Jacobi matrices and its application

YU Qian-qian, WEI Guang-sheng*   

  1. College of Mathematics and Information Science, Shaanxi Normal University, Xian 710062, Shaanxi, China
  • Received:2018-03-15 Online:2018-08-20 Published:2018-07-11

PDF (PC)

607

Abstract: The inverse spectral problem of Jacobi matrices is considered. A new inverse spectral theorem for Jacobi matrices with some rank 1 perturbations of the matrix Jn is proved. And the theorem is applied to the corresponding mass-spring system. The necessary and sufficient conditions for the existence and uniqueness of the solution are derived. Furthermore, the numerical algorithm and some numerical examples are given.

Key words: Jacobi matrices, mass-spring system, inverse spectral problem

CLC Number: 

  • O175.9
[1] KREIN M G. On the spectrum of a Jacobi matrix in connection with the torsional oscillation of shafts[J]. Matematicheskii Sbornik, 1933, 40:455-466.
[2] BORG G. Eine Unkehrung der Sturm-Liouvilleschen Eigenwert Aufgabe[J]. Acta Mathematica, 1946, 78(1):1-96.
[3] HOCHSTADT H. On the construction of a Jacobi matrix from spectral data[J]. Linear Algebra & Its Applications, 1974, 8(5):435-446.
[4] HALD O H. Inverse eigenvalue problems for Jacobi matrices[J]. Linear Algebra & its Applications, 2007, 14(1):63-85.
[5] BOLEY D, GOLUB G H. A survey of matrix inverse eigenvalue problems[J]. Inverse Problems, 1999, 3(4):595-622.
[6] 周树荃, 戴华. 代数特征值反问题[M]. 郑州:河南科学技术出版社, 1991. ZHOU Shuquan, DAI Hua. Inverse problem of algebra eigenvalue[M]. Zhengzhou: Henan Science and Technology Press, 1991.
[7] XU Shufang. On the Jacobi matrix inverse eigenvalue problem with mixed given data[J]. Siam Journal on Matrix Analysis and Applications, 1996, 17(3):632-639.
[8] LIANG Haixia, JIANG Erxiong, et al. An inverse eigenvalue problem for Jacobi matrices[J]. Computational Mathematics, 2007, 21(5):620-630.
[9] WU Xiaoqian. A divide and conquer algorithm on the double dimensional inverse eigenvalue problem for Jacobi matrices[J]. Applied Mathematics & Computation, 2012, 219(8):3840-3846.
[10] WEI Ying. Inverse eigenvalue problem of Jacobi matrix with mixed data[J]. Linear Algebra & Its Applications, 2015, 466(10):102-116.
[11] RAM Y M. Inverse eigenvalue problem for a modified vibrating system[J]. Siam Journal on Applied Mathematics, 1993, 53(6):1762-1775.
[12] NYLEN P, UHLIG F. Inverse eigenvalue problems associated with spring-mass systems[J]. Linear Algebra & its Applications, 1997, 254(1):409-425.
[13] GLADWELL G M L. Inverse problems in vibration[M]. Dordrecht: Springer, 1986.
[14] WEI Zhaoying, WEI Guangsheng. Inverse spectral problem for Jacobi matrices with partial spectral data[J]. Inverse Problems, 2011, 27(7):075007.
No related articles found!
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