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Singular Value Decomposition


If a matrix A has a matrix of eigenvectors P that is not invertible (for example, the matrix [1 1; 0 1] has the noninvertible system of eigenvectors [1 0; 0 0]), then A does not have an eigen decomposition. However, if A is an m×n real matrix with m>n, then A can be written using a so-called singular value decomposition of the form

 A=UDV^(T).
(1)

Note that there are several conflicting notational conventions in use in the literature. Press et al. (1992) define U to be an m×n matrix, D as n×n, and V as n×n. However, the Wolfram Language defines U as an m×m, D as m×n, and V as n×n. In both systems, U and V have orthogonal columns so that

 U^(T)U=I
(2)

and

 V^(T)V=I
(3)

(where the two identity matrices may have different dimensions), and D has entries only along the diagonal.

For a complex matrix A, the singular value decomposition is a decomposition into the form

 A=UDV^(H),
(4)

where U and V are unitary matrices, V^(H) is the conjugate transpose of V, and D is a diagonal matrix whose elements are the singular values of the original matrix. If A is a complex matrix, then there always exists such a decomposition with positive singular values (Golub and Van Loan 1996, pp. 70 and 73).

Singular value decomposition is implemented in the Wolfram Language as SingularValueDecomposition[m], which returns a list {U, D, V}, where U and V are matrices and D is a diagonal matrix made up of the singular values of m.


See also

Cholesky Decomposition, Eigen Decomposition, Eigen Decomposition Theorem, Eigenvalue, Eigenvector, LU Decomposition, Matrix Decomposition, QR Decomposition, Singular Value, Unitary Matrix

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References

Gentle, J. E. "Singular Value Factorization." §3.2.7 in Numerical Linear Algebra for Applications in Statistics. Berlin: Springer-Verlag, pp. 102-103, 1998.Golub, G. H. and Van Loan, C. F. "The Singular Value Decomposition" and "Unitary Matrices." §2.5.3 and 2.5.6 in Matrix Computations, 3rd ed. Baltimore, MD: Johns Hopkins University Press, pp. 70-71 and 73, 1996.Nash, J. C. "The Singular-Value Decomposition and Its Use to Solve Least-Squares Problems." Ch. 3 in Compact Numerical Methods for Computers: Linear Algebra and Function Minimisation, 2nd ed. Bristol, England: Adam Hilger, pp. 30-48, 1990.Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. "Singular Value Decomposition." §2.6 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 51-63, 1992.

Cite this as:

Weisstein, Eric W. "Singular Value Decomposition." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SingularValueDecomposition.html

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