Singular Value Decomposition and Least Squares Solutions

  • G. H. Golub
  • C. Reinsch
Part of the Die Grundlehren der mathematischen Wissenschaften book series (GL, volume 186)


Let A be a real m×n matrix with mn. It is well known (cf. [4]) that
$$A = U\sum {V^T}$$
$${U^T}U = {V^T}V = V{V^T} = {I_n}{\text{ and }}\sum {\text{ = diag(}}{\sigma _{\text{1}}}{\text{,}} \ldots {\text{,}}{\sigma _n}{\text{)}}{\text{.}}$$
The matrix U consists of n orthonormalized eigenvectors associated with the n largest eigenvalues of AA T , and the matrix V consists of the orthonormalized eigenvectors of A T A. The diagonal elements of are the non-negative square roots of the eigenvalues of A T A; they are called singular values. We shall assume that
$${\sigma _1} \geqq {\sigma _2} \geqq \cdots \geqq {\sigma _n} \geqq 0.$$
Thus if rank(A)=r, σ r+1 = σ r+2=⋯=σ n = 0. The decomposition (1) is called the singular value decomposition (SVD).


Linear Algebra Matrix Versus Machine Precision Bell Telephone Laboratory Householder Transformation 
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.
    Businger, P., Golub, G.: Linear least squares solutions by Householder transformations. Numer. Math. 7, 269 -276 (1965). Cf. I/8.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Forsythe, G.E., Henrici, P.: The cyclic Jacobi method for computing the principal values of a complex matrix. Proc. Amer. Math. Soc. 94, 1 -23 (I960).MathSciNetzbMATHGoogle Scholar
  3. 3.
    Forsythe, G.E., Henrici, P.Golub, G.: On the stationary values of a second-degree polynomial on the unit sphere. J. Soc. Indust. Appl. Math. 13, 1050–1068 (1965).MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Forsythe, G.E., Henrici, P.Moler, C. B.: Computer solution of linear algebraic systems. Englewood Cliffs, New Jersey: Prentice-Hall 1967.zbMATHGoogle Scholar
  5. 5.
    Francis, J.: The Q R transformation. A unitary analogue to the L R transformation. Comput. J. 4, 265–271 (1961, 1962).MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Golub, G., Kahan, W.: Calculating the singular values and pseudo-inverse of a matrix. J. SIAM. Numer. Anal., Ser. B 2, 205 -224 (1965).MathSciNetGoogle Scholar
  7. 7.
    Golub, G., Kahan, W. Least squares, singular values, and matrix approximations. Aplikace Matematiky 13, 44–51 (1968).MathSciNetzbMATHGoogle Scholar
  8. 8.
    Hestenes, M. R.: Inversion of matrices by biorthogonalization and related results. J. Soc. Indust. Appl. Math. 6, 51–90 (1958).MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Kogbetliantz, E. G.: Solution of linear equations by diagonalization of coefficients matrix. Quart. Appl. Math. 13, 123 -132 (1955).MathSciNetzbMATHGoogle Scholar
  10. 10.
    Kublanovskaja, V.N.: Some algorithms for the solution of the complete problem of eigenvalues.Ž . Vyisl. Mat. i Mat. Fiz. 1, 555 -570 (1961).MathSciNetGoogle Scholar
  11. 11.
    Martin, R. S., Reinsch, C., Wilkinson, J.H.: Householder’s tridiagonalization of a symmetric matrix. Numer. Math. 11, 181 -195 (1968). Cf. II/2.MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Wilkinson, J.: Error analysis of transformations based on the use of matrices of the form I -2w wH. Error in digital computation, vol. II, L.B. Rail, ed., p. 77 -101. New York: John Wiley & Sons, Inc. 1965.Google Scholar
  13. 13.
    Wilkinson, J Global convergence of tridiagonal QR algorithm with origin shifts. Lin. Alg. and its Appl. 1, 409–420 (1968).MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin · Heidelberg 1971

Authors and Affiliations

  • G. H. Golub
  • C. Reinsch

There are no affiliations available

Personalised recommendations