Singular value decomposition

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In linear algebra, the singular-value decomposition (SVD) is a factorization of a real or complex matrix. It is the generalization of the eigendecomposition of a positive semidefinite normal matrix (for example, a symmetric matrixwith positive eigenvalues) to any  matrix via an extension of the polar decomposition. It has many useful applications in signal processing and statistics.

Formally, the singular-value decomposition of an  real or complex matrix  is a factorization of the form , where  is an  real or complex unitary matrix,  is an  rectangular diagonal matrix with non-negative real numbers on the diagonal, and  is an  real or complex unitary matrix. The diagonal entries  of  are known as the singular values of . The columns of  and the columns of  are called the left-singular vectors and right-singular vectors of , respectively.

The singular-value decomposition can be computed using the following observations:

  • The left-singular vectors of M are a set of orthonormal eigenvectors of MM.
  • The right-singular vectors of M are a set of orthonormal eigenvectors of MM.
  • The non-zero singular values of M (found on the diagonal entries of Σ) are the square roots of the non-zero eigenvalues of both MM and MM.