Concept# Singular value decomposition

Summary

In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix. It generalizes the eigendecomposition of a square normal matrix with an orthonormal eigenbasis to any \ m \times n\ matrix. It is related to the polar decomposition.
Specifically, the singular value decomposition of an \ m \times n\ complex matrix M is a factorization of the form \ \mathbf{M} = \mathbf{U\Sigma V^*}\ , where U is an \ m \times m\ complex unitary matrix, \ \mathbf{\Sigma}\ is an \ m \times n\ rectangular diagonal matrix with non-negative real numbers on the diagonal, V is an n \times n complex unitary matrix, and \ \mathbf{V^*}\ is the conjugate transpose of V. Such decomposition always exists for any complex matrix. If M is real, then U and V can be guaranteed to be real orthogonal matrices; in such contexts, the SVD is often denoted \ \mathbf{ U\S

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