, Volume 58, Issue 1, pp 115–118 | Cite as

An alternating least squares method for the weighted approximation of a symmetric matrix

  • Jos M. F. Berge
  • Henk A. L. Kiers


Bailey and Gower examined the least squares approximationC to a symmetric matrixB, when the squared discrepancies for diagonal elements receive specific nonunit weights. They focussed on mathematical properties of the optimalC, in constrained and unconstrained cases, rather than on how to obtainC for any givenB. In the present paper a computational solution is given for the case whereC is constrained to be positive semidefinite and of a fixed rankr or less. The solution is based on weakly constrained linear regression analysis.

Key words

matrix approximation weakly constrained regression 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Bailey, R. A., & Gower, J. C. (1990). Approximating a symmetric matrix.Psychometrika, 55, 665–675.Google Scholar
  2. Browne, M. W. (1987). The Young-Householder algorithm and the least-squares multidimensional scaling of squared distances.Journal of Classification, 4, 175–190.Google Scholar
  3. Eckart, C., & Young, G. (1936). The approximation of one matrix by another of lower rank.Psychometrika, 1, 211–218.Google Scholar
  4. Gabriel, K. R., & Zamir, S. (1979). Lower rank approximation of matrices by least squares with any choice of weights.Technometrics, 21, 489–498.Google Scholar
  5. Golub, G. H., & Von Matt, U. (1991). Quadratically constrained least squares and quadratic problems.Numerische Mathematik, 59, 561–580.Google Scholar
  6. Harman, H. H., & Jones, W. H. (1966). Factor analysis by minimizing residuals (minres).Psychometrika, 31, 351–368.Google Scholar
  7. Knol, D. L., & ten Berge, J. M. F. (1989). Least-squares approximation of an improper correlation matrix by a proper one.Psychometrika, 54, 53–61.Google Scholar
  8. Nosal, M. (1977). A note on the MINRES method.Psychometrika, 42, 149–151.Google Scholar
  9. ten Berge, J. M. F. (1991). A general solution for a class of weakly constrained linear regression problems.Psychometrika, 56, 601–609.Google Scholar
  10. ten Berge, J. M. F., & Zegers, F. E. (1990). Convergence properties of certain MINRES algorithms.Multivariate Behavioral Research, 25, 421–425.Google Scholar

Copyright information

© The Psychometric Society 1993

Authors and Affiliations

  • Jos M. F. Berge
    • 1
  • Henk A. L. Kiers
    • 1
  1. 1.Department of PsychologyUniversity of GroningenGroningenThe Netherlands

Personalised recommendations