The Jacobi Method for Real Symmetric Matrices

  • H. Rutishauser
Part of the Die Grundlehren der mathematischen Wissenschaften book series (GL, volume 186)


As is well known, a real symmetric matrix can be transformed iteratively into diagonal form through a sequence of appropriately chosen elementary orthogonal transformations (in the following called Jacobi rotations):
$${A_k} \to {A_{k + 1}} = U_k^T{A_k}{U_k}{\text{ (}}{A_0}{\text{ = given matrix),}}$$
where U k = U k(p,q, φ) is an orthogonal matrix which deviates from the unit matrix only in the elements
$${u_{pp}} = {u_{qq}} = \cos (\varphi ){\text{ and }}{u_{pq}} = - {u_{qp}} = \sin (\varphi ).$$


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Barth, W., R. S. Martin, and J. H. Wilkinson: Calculation of the eigenvectors of a symmetric tridiagonal matrix by the method of bisection. Numer. Math. 9, 386–393 (1967). Cf. II/S.MathSciNetCrossRefGoogle Scholar
  2. [2]
    Gregory, R. T.: Computing eigenvalues and eigenvectors of a symmetric matrix on the ILLIAC. Math. Tab. and other Aids to Comp. 7, 215–220 (1953).MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    Henrici, P.: On the speed of convergence of cyclic and quasicyclic Jacobi methods for computing eigenvalues of Hermitian matrices. J. Soc. Indust. Appl. Math. 6, 144–162 (1958).MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    Jacobi, C. G. J.: Über ein leichtes Verfahren, die in der Theorie der Säkularstörungen vorkommenden Gleichungen numerisch aufzulösen. Crelle’s Journal 30, 51–94 (1846).zbMATHCrossRefGoogle Scholar
  5. [5]
    Pope, D. A., and C. Tompkins. Maximizing functions of rotations-experiments concerning speed of diagonalisation of symmetric matrices using Jacobi’s method. J. Assoc. Comput. Mach. 4, 459–466 (1957).MathSciNetCrossRefGoogle Scholar
  6. [6]
    Rutishauser, H., and H. R. Schwarz. The LR-transformation method for symmetric matrices. Numer. Math. 5, 273–289 (1963).MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    Schoenhage, A.: Zur Konvergenz des Jacobi-Verfahrens. Numer. Math. 3, 374– 380 (1961).MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    Wilkinson, J. H.: Note on the quadratic convergence of the cyclic Jacobi proessess . Numer. Math. 4, 296–300 (1962).MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    Wilkinson, J. H The algebraic eigenvalue problem, 662 p. Oxford: Clarendon Press 1965.Google Scholar
  10. [10]
    Martin, R. S., C. Reinsch, and J. H. Wilkinson. Householder’s tridiagonaliza-tion of a symmetric matrix. Numer. Math. 11, 181–195 (1968). Cf. II/2.MathSciNetzbMATHCrossRefGoogle Scholar
  11. [11]
    Bowdler, H., R. S. Martin, C. Reinsch, and J. H. Wilkinson. The QR and QL algorithms for symmetric matrices. Numer. Math. 11, 293 -306 (1968). Cf. II/3.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    Dubrulle, A., R. S. Martin, and J. H. Wilkinson. The implicit QL algorithm. Numer. Math. 12, 377–383 (1968). Cf. II/4.MathSciNetCrossRefGoogle Scholar
  13. [13]
    Peters, G., and J. H. Wilkinson. The calculation of specified eigenvectors by inverse iteration. Cf. II/18.Google Scholar

Copyright information

© Springer-Verlag Berlin · Heidelberg 1971

Authors and Affiliations

  • H. Rutishauser

There are no affiliations available

Personalised recommendations