BUILT-IN MATHEMATICA SYMBOL

SingularValueDecomposition

SingularValueDecomposition[m]
gives the singular value decomposition for a numerical matrix m as a list of matrices

, where w is a diagonal matrix and m can be written as u.w.Conjugate[Transpose[v]].

gives the generalized singular value decomposition of m with respect to a.

SingularValueDecomposition

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

SingularValueDecomposition

gives the generalized singular value decomposition associated with the k largest singular values.

MORE INFORMATION

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 gives a list of matrices such that m can be written as u.w.Conjugate[Transpose[v]] and a can be written as ua.wa.Conjugate[Transpose[v]]. »

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Scope (3)

m is a 3×2 matrix:

Find the singular value decomposition of m using machine-number arithmetic:

Find the singular value decomposition of m using 24-digit precision arithmetic:

The singular value decomposition of a random complex-valued 2×4 matrix:

m and a are random matrices with 3 columns:

Find the generalized singular value decomposition of m with respect to a:

Generalizations & Extensions (2)

m is a singular 3×3 matrix:

Find the full singular value decomposition of m:

The original matrix can be reconstructed from the singular value decomposition:

Find the "thin" decomposition associated with the nonzero singular values:

This decomposition still has sufficient information to reconstruct the matrix:

s is a large sparse matrix:

Options (1)

m is a nearly singular matrix:

To machine precision, the matrix is effectively singular:

With a smaller tolerance, the nonzero singular value is detected:

The default tolerance is based on precision, so the small value is detected with precision 20:

Applications (2)

m is a 2×2 matrix:

Find its singular value decomposition:

The columns of v give the directions of minimal and maximal stretching of vectors by

:

The columns of u give the directions of minimal and maximal stretching of vectors by

:

Here is some randomly generated data:

Construct a design matrix for fitting the data to basis functions

:

Find the condensed singular value decomposition:

Find a vector x that minimizes

:

The components of x are the coefficients given by Fit:

Properties & Relations (4)

m and a are random matrices with 4 columns:

Find the singular value decomposition of m:

Verify that m is equal to u.w.Conjugate[Transpose[v]]:

Find the generalized singular value decomposition of m with respect to a:

Verify that m is equal to u.w.Conjugate[Transpose[v]]:

Verify that a is equal to ua.wa.Conjugate[Transpose[v]]:

m is a random 2×5 matrix:

Find the singular value decomposition of m:

The diagonal elements of w are the square roots of the eigenvalues of m.Transpose[m]:

The columns of u are the eigenvectors of m.Transpose[m] up to sign:

The first two columns of v are the eigenvectors of Transpose[m].m up to sign:

m is a 3×3 singular matrix:

Find the thin singular value decomposition:

Form the inverse of the diagonal matrix w:

Construct the Moore-Penrose pseudoinverse of m:

This is the matrix given by the PseudoInverse command:

m is the outer product of two vectors:

The condensed singular value decomposition for m:

The single column of u and v are normalizations of the two vectors:

The element of w is the product of the norms:

Possible Issues (1)

m is a 2×1000 random matrix:

The full singular value decomposition is very large because u is a 1000×1000 matrix:

The condensed singular value decomposition is much smaller:

It still contains sufficient information to reconstruct m: