A Constrained Eigenvalue Problem

  • Walter Gander
  • Gene H. Golub
  • Urs von Matt
Part of the NATO ASI Series book series (volume 70)


In this paper we consider the following mathematical and computational problem. Given the quantities
  • A: (n + m)-by-(n + m) matrix, symmetric, n > 0

  • N: (n + m)-by-m matrix with full rank

  • t: vector of dimension m with ∥(NT)+t∥ < 1

Determine an x such that
$$ {\operatorname{x} ^T}Ax = \min $$
subject to the constraints
$$ {N^T}x = t $$
$$ {x^T}x = 1. $$
Variants of this problem occur in many applications [1,5,7,8,11]. The problem has been studied previously when t = 0, the null vector, (cf. [4,6]). When t ≠ 0, then the problem becomes more complicated.

We show how to eliminate the linear constraint (i). Then three different methods are presented for the solution of the resulting Lagrange equations.


Eigenvalue Problem Linear Constraint Full Rank Lagrange Equation Null Vector 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    N. R. Draper, “Ridge Analysis” of Response Surfaces, Technometrics, 5 (1963), pp. 469–479.MATHCrossRefGoogle Scholar
  2. [2]
    G. E. Forsythe and G. H. Golub, On the stationary values of a second-degree polynomial on the unit sphere, SIAM J. Appl. Math., 13 (1965), pp. 1050–1068.MathSciNetMATHGoogle Scholar
  3. [3]
    W. Gander, Least Squares with a Quadratic Constraint,Numer. Math., 36 (1981), pp. 291–307.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    G. H. Golub, Some Modified Matrix Eigenvalue Problems, SIAM Review, 15 (1973), pp. 318–334.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    G. H. Golub, M. Heath and G. Wahba, Generalized Cross-Validation as a Method for Choosing a Good Ridge Parameter,Technometrics, 21 (1979), pp. 215–223.MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    G. H. Golub and C. F. van Loan, Matrix Computations, The Johns Hopkins University Press, Baltimore, 1983.MATHGoogle Scholar
  7. [7]
    J. J. Moré, The Levenberg-Marquardt Algorithm: Implementation and Theory, in Proc. of the Biennial Conference held at Dundee, ed. A. Dold and B. Eckmann, Springer-Verlag, 1978, pp. 105–116.Google Scholar
  8. [8]
    J. J. Moré and D. C. Sorensen, Computing a Trust Region Step, SIAM J. Sci. Stat. Comput., 4 (1983), pp. 553–572.MATHCrossRefGoogle Scholar
  9. [9]
    Chr. H. Reinsch, Smoothing by Spline Functions. II, Numer. Math., 16 (1971), pp. 451–454.MathSciNetCrossRefGoogle Scholar
  10. [10]
    E. Spjotvoll, A Note on a Theorem of Forsythe and Golub, SIAM J. Appl. Math., 23 (1972), pp. 307–311.MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    E. Spjbtvoll, Multiple Comparison of Regression Functions,Ann. Math. Statist., 43 (1972), pp. 1076–1088.MathSciNetCrossRefGoogle Scholar
  12. [12]
    U. von Matt, A Constrained Eigenvalue Problem, Diploma Thesis, Abteilung für Informatik, ETH Zürich, 1988.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1991

Authors and Affiliations

  • Walter Gander
    • 1
  • Gene H. Golub
    • 2
  • Urs von Matt
    • 1
  1. 1.Institut für InformatikETH ZentrumZürichSwitzerland
  2. 2.Department of Computer ScienceStanford UniversityStanfordUSA

Personalised recommendations