Matrix Inverse Eigenvalue Problems

  • Graham M. L. Gladwell
Part of the CISM International Centre for Mechanical Sciences book series (CISM, volume 529)


The term matrix eigenvalue problems refers to the computation of the eigenvalues of a symmetric matrix. By contrast, the term inverse matrix eigenvalue problem refers to the construction of a symmetric matrix from its eigenvalues. While matrix eigenvalue problems are well posed, inverse matrix eigenvalue problems are ill posed: there is an infinite family of symmetric matrices with given eigenvalues. This means that either some extra constraints must be imposed on the matrix, or some extra information must be supplied. These lectures cover four main areas: i) Classical inverse problems relating to the construction of a tridiagonal matrix from its eigenvalues and the first (or last) components of its eigenvectors. ii) Application of these results to the construction of simple in-line mass-spring systems, and a discussion of extensions of these results to systems with tree structure. iii) Isospectral systems (systems that all have the same eigenvalues) studied in the context of the QR algorithm, with special attention paid to the important concept of total positivity. iv) Introduction to the concept of Toda flow, a particular isospectral flow.


Inverse Problem Eigenvalue Problem Jacobi Matrix Tridiagonal Matrix Total Positivity 
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  1. S.A. Andrea and T.G. Berry. Continued fractions and periodic Jacobi matrices. Linear Algebra and Its Applications, 161:117–134, 1992.MathSciNetzbMATHCrossRefGoogle Scholar
  2. P. Arbenz and G.H. Golub. Matrix shapes invariant under the symmetric QR algorithm. Numerical Linear Algebra with Applications, 2:87–93, 1995.MathSciNetzbMATHCrossRefGoogle Scholar
  3. A. Berman and R.J. Plemmons. Nonnegative matrices in the mathematical sciences. Society for Industrial and Applied Mathematics, 1994.Google Scholar
  4. D. Boley and G.H. Golub. A modified method for reconstructing periodic Jacobi matrices. Mathematics of Computation, 42:143–150, 1984.MathSciNetzbMATHCrossRefGoogle Scholar
  5. D. Boley and G.H. Golub. A survey of matrix inverse eigenvalue problems. Inverse Problems, 3:595–622, 1987.MathSciNetzbMATHCrossRefGoogle Scholar
  6. M.T. Chu. The generalized Toda flow, the QR algorithm and the center manifold theory. SIAM Journal on Algebraic and Discrete Methods, 5: 187–201, 1984.zbMATHCrossRefGoogle Scholar
  7. M.T. Chu and G.H. Golub. Inverse Eigenvalue Problems: Theory, Algorithms and Applications. Oxford University Press, 2005.Google Scholar
  8. A.L. Duarte. Construction of acyclic matrices from spectral data. Linear Algebra and Its Applications, 113:173–182, 1989.MathSciNetzbMATHCrossRefGoogle Scholar
  9. S. Friedland. Inverse eigenvalue problems. Linear Algebra and Its Applications, 17:15–51, 1977.MathSciNetzbMATHCrossRefGoogle Scholar
  10. G.M.L. Gladwell. Inverse Problems in Vibration. Kluwer Academic Publishers, 2004.Google Scholar
  11. G.M.L. Gladwell. Minimal mass solutions to inverse eigenvalue problems. Inverse Problems, 22:539–551, 2006.MathSciNetzbMATHCrossRefGoogle Scholar
  12. G.M.L. Gladwell. The inverse problem for the vibrating beam. Proceedings of the Royal Society of London — Series A, 393:277–295, 1984.MathSciNetzbMATHCrossRefGoogle Scholar
  13. G.M.L. Gladwell. Total positivity and the QR algorithm. Linear Algebra and Its Applications, 271:257–272, 1998.MathSciNetzbMATHCrossRefGoogle Scholar
  14. G.M.L. Gladwell and O. Rojo. Isospectral flows that preserve matrix structure. Linear Algebra and Its Applications, 421:85–96, 2007.MathSciNetzbMATHCrossRefGoogle Scholar
  15. G.M.L. Gladwell, K.R. Lawrence, and D. Siegel. Isospectral finite element membranes. Mechanical Systems and Signal Processing, 23:1986–1999, 2009.CrossRefGoogle Scholar
  16. G.H. Golub. Some uses of the Lanczos algorithm in numerical linear algebra. In J.H.H. Miller, editor, Topics in Numerical Analysis. Academic Press, 1973.Google Scholar
  17. T. Nanda. Differential equations and the QR algorithm. SIAM Journal of Numerical Analysis, 22:310–321, 1985.MathSciNetzbMATHCrossRefGoogle Scholar
  18. P. Nylen and F. Uhlig. Realizations of interlacing by tree-patterned matrices. Linear Multilinear Algebra, 38:13–37, 1994.MathSciNetzbMATHCrossRefGoogle Scholar
  19. Lord Rayleigh. The Theory of Sound. Macmillan, 1894.Google Scholar
  20. M. Toda. Waves in nonlinear lattices. Progress of Theoretical Physicis Supplement, 45:174–200, 1970.CrossRefGoogle Scholar
  21. D.K. Vijay. Some Inverse Problems in Mechanics. University of Waterloo, M.A.Sc. Thesis, 1972.Google Scholar
  22. S.F. Xu. An Introduction to Inverse Algebraic Eigenvalue Problems. Vieweg, 1998.Google Scholar

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© CISM, Udine 2011

Authors and Affiliations

  • Graham M. L. Gladwell
    • 1
  1. 1.Department of Civil EngineeringUniversity of WaterlooWaterlooCanada

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