# Wolfram Language & System 11.0 (2016)|Legacy Documentation

This is documentation for an earlier version of the Wolfram Language.
BUILT-IN WOLFRAM LANGUAGE SYMBOL

# SingularValueDecomposition

gives the singular value decomposition for a numerical matrix m as a list of matrices {u,w,v}, where w is a diagonal matrix and m can be written as u.w.Conjugate[Transpose[v]].

SingularValueDecomposition[{m,a}]
gives the generalized singular value decomposition of m with respect to a.

gives the singular value decomposition associated with the k largest singular values of m.

gives the decomposition for the k largest singular values, or as many as are available.

SingularValueDecomposition[{m,a},k]
gives the generalized singular value decomposition associated with the k largest singular values.

## Details and OptionsDetails and Options

• The matrix m may be rectangular.
• The diagonal elements of w are the singular values of m.
• SingularValueDecomposition sets to zero any singular values that would be dropped by SingularValueList.
• The option Tolerance can be used as in SingularValueList to determine which singular values will be considered to be zero. »
• u and v are column orthonormal matrices, whose transposes can be considered as lists of orthonormal vectors.
• SingularValueDecomposition[{m,a}] gives a list of matrices {{u,ua},{w,wa},v} such that m can be written as u.w.Conjugate[Transpose[v]] and a can be written as ua.wa.Conjugate[Transpose[v]]. »
• gives the decomposition for the k largest singular values, or as many as are available.

## ExamplesExamplesopen allclose all

### Basic Examples  (1)Basic Examples  (1)

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