Numerische Mathematik

, Volume 98, Issue 1, pp 167–176 | Cite as

On sufficient and necessary conditions for the Jacobi matrix inverse eigenvalue problem

  • Linzhang Lu
  • Michael K.  Ng


In this paper, we study the inverse eigenvalue problem of a specially structured Jacobi matrix, which arises from the discretization of the differential equation governing the axial of a rod with varying cross section (Ram and Elhay 1998 Commum. Numer. Methods Engng. 14 597-608). We give a sufficient and some necessary conditions for such inverse eigenvalue problem to have solutions. Based on these results, a simple method for the reconstruction of a Jacobi matrix from eigenvalues is developed. Numerical examples are given to demonstrate our results.


Differential Equation Eigenvalue Problem Jacobi Matrix Vary Cross Section Inverse Eigenvalue Problem 
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  1. 1.
    Boley, D., Golub, G.H: A survey of matrix inverse eigenvalue problem. Inverse Problem 3, 595–622 (1987)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Chu, M.T.: Inverse eigenvalue problems SIAM Rev. 40, 1–39 (1998)Google Scholar
  3. 3.
    Gantmacher, F.R.: The Theory of Matrices (New York: Chelsea) 1959Google Scholar
  4. 4.
    Gladwell, G.M.L.: Inverse Problem in Vibration (Dordrecht: Nijhoff) 1986Google Scholar
  5. 5.
    Golub, G.H., Van Loan, C.: Matrix Computations (London: Academic) 1983Google Scholar
  6. 6.
    Lu, L.Z., Sun, W. W.: On necessary conditions for reconstruction of a specially structured Jacobi matrix from eigenvalues. Inverse Problem 15, 977–987 (1999)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Osborne, M.R.: On the Inverse Eigenvalue Problem for Matrices and Related Problems for Difference and Differential Equations(Lecture Notes in Mathematics vol 228) (New York: Springer) 155–168 (1971)Google Scholar
  8. 8.
    Ram, Y.M.: An inverse eigenvalue problem for modified vibrating system SIAM J Appl Math 53, 1762–1775 (1993)Google Scholar
  9. 9.
    Ram, Y.M., Elhay, S.: Constructing the shape of a rod from eigenvalues Commun. Numer. Methods Engng 14, 597–608 (1998)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.Department of MathematicsXiamen UniversityXiamenPeople’s Republic of China
  2. 2.Department of MathematicsThe University of Hong KongHong Kong

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