Math 2.0

Linear and multilinear algebra; matrix theory Canonical forms and decompositions Theorem

Singular value decomposition

Builds on Singular valueUnitary matrix

Statement

Let AA be an m×nm \times n complex matrix. Then there are a unitary matrix UU of size m×mm \times m, a unitary matrix VV of size n×nn \times n, and an m×nm \times n matrix Σ\Sigma with A=UΣV∗.A = U \Sigma V^*. Here Σij=0\Sigma_{ij} = 0 for i≠ji \ne j, and Σ11≥Σ22≥⋯≥0\Sigma_{11} \ge \Sigma_{22} \ge \cdots \ge 0 are the singular values of AA.

Remarks

If AA is real, then UU and VV can be chosen real orthogonal. The columns of VV are eigenvectors of A∗AA^* A, and the columns of UU are eigenvectors of AA∗A A^*. The matrix Σ\Sigma is unique; UU and VV are not unique in general. Keeping the kk largest singular values gives the best approximation of AA of rank kk, for the Frobenius and operator norms.

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Identity matrix, object, no evidence levelIdentity matrixobjectMatrix, definition, no evidence levelMatrixdefinitionVector space, definition, no evidence levelVector spacedefinitionSquare matrix, definition, no evidence levelSquare matrixdefinitionTranspose, definition, no evidence levelTransposedefinitionLinear map, definition, no evidence levelLinear mapdefinitionConjugate transpose, definition, no evidence levelConjugate transposedefinitionEndomorphism, definition, no evidence levelEndomorphismdefinitionHermitian matrix, definition, no evidence levelHermitian matrixdefinitionUnitary matrix, definition, no evidence levelUnitary matrixdefinitionEigenvalue, definition, no evidence levelEigenvaluedefinitionPositive semidefinite matrix, definition, no evidence levelPositive semidefinite…definitionGram matrices are positive semidefinite, statement, no evidence levelGram matrices are pos…statementSingular value, definition, no evidence levelSingular valuedefinitionSingular value decomposition, statement, no evidence levelSingular value decomp…statement
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  1. Identity matrix object
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  13. Gram matrices are positive semidefinite statement
  14. Singular value definition
  15. Singular value decomposition statement

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Further reading

  1. Web https://en.wikipedia.org/wiki/Singular_value_decomposition, lead section
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Tag000NPTMX6J
URIm2://000NPTMX6J
Kindstatement, theorem
WritingW1 stated
MSC15A18, 15A23
StatusThis copy of the site has no build data. Run m2 status 000NPTMX6J.
UsesSingular value, Unitary matrix
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