Dictionary of Applied Machine Learning

singular value decomposition

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The singular value decomposition (SVD) is a factorization of a matrix $\mA \in \reals^{\samplesize \times \dimlocalmodel}$ of the form $\mA = \mV \bm{\Lambda} \mU^{\top}$ with orthonormal matrices $\mV$ and $\mU$. The matrix $\bm{\Lambda}$ is nonzero only along its main diagonal. The diagonal entries of $\bm{\Lambda}$ are nonnegative and referred to as singular values. In contrast to an eigenvalue decomposition (EVD), which requires a square diagonalizable matrix, an SVD exists for every matrix.

Definition

The SVD for a matrix $\mA \in \mathbb{R}^{\samplesize \times \dimlocalmodel}$ is a factorization of the following form: \[\mA = \mathbf{V} {\bm \Lambda} \mathbf{U}^{\top}\] with orthonormal matrices \[\mV = \big(\vv^{(1)},\,\ldots,\,\vv^{(\samplesize)}\big) \in \mathbb{R}^{\samplesize \times \samplesize}, \qquad \mU = \big( \vu^{(1)},\,\ldots,\,\vu^{(\dimlocalmodel)} \big) \in \mathbb{R}^{\dimlocalmodel \times \dimlocalmodel}\] (Golub and Loan, 2013) (see Fig.\ 1).

Figure 1 of the entry svd
Figure 1: Orthonormal matrices $\mV$ and $\mU$
The matrix ${\bm \Lambda} \in \mathbb{R}^{\samplesize \times \dimlocalmodel}$ is only nonzero along the main diagonal, whose entries $\Lambda_{\featureidx,\featureidx}$ are nonnegative and referred to as singular values.

See also: matrix.

References

  1. Golub and Loan (2013). Matrix Computations. The Johns Hopkins Univ. Press.

Cite this entry

@misc{dictml_svd,
  author = {Jung, Alexander},
  title = {singular value decomposition},
  howpublished = {Dictionary of Applied Machine Learning (course edition)},
  year = {2026},
  doi = {10.5281/zenodo.21569296},
  note = {ISBN 978-952-64-3013-3, CC BY 4.0, retrieved 2026-08-06},
  url = {https://dictionaryofml.org/terms/svd.html}
}