Title: singular value decomposition
Long Title: IUPAC Gold Book - singular value decomposition
DOI: 10.1351/goldbook.10115
Status: current

Definition
A factorization of an \(m \times n\) matrix (\(\boldsymbol{M}\)) such that \(\boldsymbol{M} = \boldsymbol{U}\boldsymbol{\Sigma} \boldsymbol{V}^{T}\), where \(\boldsymbol{U}\) is an \(m \times m\) matrix, \(\boldsymbol{\Sigma}\) is a \(m \times n\) matrix and \(\boldsymbol{V}^{\rm{T}}\) is a \(n \times n\) matrix.

Note
If \(\boldsymbol{M}\) is a data matrix with \(m\) objects and \(n\) variables, the matrix \(\boldsymbol{U}\) is the scores matrix, the diagonal of \(\boldsymbol{\Sigma}\) contain the square roots of the eigenvalues and \(\boldsymbol{V}\) is the loadings matrix.

Related Terms
- data matrix: https://goldbook.iupac.org//terms/view/10047
- loadings: https://goldbook.iupac.org//terms/view/10096
- scores: https://goldbook.iupac.org//terms/view/10112

Source
- PAC, 2016, 88, 407. 'Vocabulary of concepts and terms in chemometrics (IUPAC Recommendations 2016)' on page 425 (https://doi.org/10.1515/pac-2015-0605)

Other Outputs
- html: https://goldbook.iupac.org/terms/view/10115/html
- json: https://goldbook.iupac.org/terms/view/10115/json
- xml: https://goldbook.iupac.org/terms/view/10115/xml

Citation: Citation: 'singular value decomposition' in IUPAC Compendium of Chemical Terminology, 5th ed. International Union of Pure and Applied Chemistry; 2025. Online version 5.0.0, 2025. 10.1351/goldbook.10115
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Accessed: 2026-07-27T04:51:28+00:00