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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)

Abstract

Let A be a real m×n matrix with mn. It is well known (cf. [4]) that
$$A = U\sum {V^T}$$
(1)
where
$${U^T}U = {V^T}V = V{V^T} = {I_n}{\text{ and }}\sum {\text{ = diag(}}{\sigma _{\text{1}}}{\text{,}} \ldots {\text{,}}{\sigma _n}{\text{)}}{\text{.}}$$
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).

Keywords

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.

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© Springer-Verlag Berlin · Heidelberg 1971

Authors and Affiliations

  • G. H. Golub
  • C. Reinsch

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