Skip to main content

Rank-one modification of the symmetric eigenproblem


An algorithm is presented for computing the eigensystem of the rank-one modification of a symmetric matrix with known eigensystem. The explicit computation of the updated eigenvectors and the treatment of multiple eigenvalues are discussed. The sensitivity of the computed eigenvectors to errors in the updated eigenvalues is shown by a perturbation analysis.

This is a preview of subscription content, access via your institution.


  1. Bunch, J.R., Nielsen, C.P.: Updating the singular value decomposition. Numer. Math. (to appear)

  2. Bunch, J.R., Rose, D.J.: Partitioning, tearing and modification of sparse linear systems. J. Math. Anal. Appl.48, 574–593 (1074)

    Google Scholar 

  3. Dennis, J.E., Moré, J.J.: Quasi-Newton methods, motivation and theory. SIAM Rev.19, 46–89 (1977)

    Google Scholar 

  4. Golub, G.H.: Some modified matrix eigenvalue problems. SIAM Rev.15, 318–334 (1973)

    Google Scholar 

  5. Stewart, G.W.: Introduction to matrix computations. New York: Academic Press 1973

    Google Scholar 

  6. Thompson, R.C.: The behavior of eigenvalues and singular values under perturbations of restricted rank. Linear Algebra and Appl.13, 69–78 (1976)

    Google Scholar 

  7. Wilkinson, J.H.: The algebraic eigenvalue problem. Oxford: Clarendon Press 1965

    Google Scholar 

Download references

Author information

Authors and Affiliations


Additional information

Support for this research was provided by NSF grants MCS 75-06510 and MCS 76-03139

Support for this research was provided by the Applied Mathematics Division, Argonne National Laboratory, Argonne, IL 60439, USA

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Bunch, J.R., Nielsen, C.P. & Sorensen, D.C. Rank-one modification of the symmetric eigenproblem. Numer. Math. 31, 31–48 (1978).

Download citation

  • Received:

  • Issue Date:

  • DOI:

Subject Classifications

  • AMS(MOS): 65F15
  • CR: 5.14