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SingularValueList

SingularValueList[m]

gives a list of the nonzero singular values of a matrix m.

SingularValueList[{m,a}]

gives the generalized singular values of m with respect to a.

SingularValueList[m,k]

gives the k largest singular values of m.

SingularValueList[{m,a},k]

gives the k largest generalized singular values of m.

Details and Options

Singular values are sorted from largest to smallest.
Repeated singular values appear with their appropriate multiplicity.
By default, singular values are kept only when they are larger than 100 times 10^-p, where p is Precision[m].
SingularValueList[m,Tolerance->t] keeps only singular values that are at least t times the largest singular value.
SingularValueList[m,Tolerance->0] returns all singular values.
The matrix m can be rectangular; the total number of singular values is always Min[Dimensions[m]].
Exact and symbolic matrices can be used, with zero tolerance assumed by default.
The singular values can be obtained from Sqrt[Eigenvalues[ConjugateTranspose[m].m]].

Examples

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Basic Examples (2)

Compute the singular values of an invertible matrix:

Compute the nonzero singular values of a singular matrix:

There are only two, rather than three, because that is the rank of the matrix:

Scope (19)

Basic Uses (7)

Find the nonzero singular values of a machine-precision matrix:

Singular values of a complex matrix:

Singular values for an exact matrix:

Singular values for an arbitrary-precision matrix:

Singular values of a symbolic matrix:

The singular values of large numerical matrices are computed efficiently:

Singular values of a non-square matrix:

Subsets of Singular Values (5)

Find the three largest singular values:

And the three smallest singular values:

Find the four largest singular values, including zero values, or as many as there are if fewer:

Zero singular values are included when computing the smallest singular values:

Repeated singular values are listed multiple times:

Repeated singular values are counted separately when extracting a subset of the singular values:

Generalized Singular Values (3)

Generalized machine-precision singular values:

Find the two smallest generalized singular values:

Find the generalized singular values of a machine-precision complex matrix:

Special Matrices (4)

Singular values of sparse matrices:

Find the three largest singular values:

Singular values of structured matrices:

Use a different structure:

The units go with the singular values:

IdentityMatrix always has all-one singular values:

Singular values of HilbertMatrix:

Options (2)

Tolerance (2)

Compute the singular values larger than of the largest singular value:

Setting Tolerance to will directly compute the same set of singular values:

m is a 16×16 Hilbert matrix:

The matrix is positive definite, so with exact arithmetic there are 16 nonzero singular values:

Many of the singular values are too small to show up at machine precision:

Setting the tolerance to zero will make them all show up:

Because of numerical roundoff, the values are not computed accurately:

Applications (4)

Find the maximum value of for the following matrix :

The maximum is equal to the largest singular value of :

Confirm the result using MaxValue:

Find the minimum value of subject to for the following matrix :

Because the matrix has at least as many rows as columns, the minimum is the smallest singular value of :

Confirm the result using MinValue:

The operator norm of a matrix, also called the spectral norm or 2-norm, is defined as the maximum value of subject to the constraint that . Find the operator norm of the following :

The maximum of , and thus the norm, is the largest singular value of :

Verify the result using Norm:

Use the singular values of to compute its norm, the norm of and the condition number of the matrices:

Find the singular values of :

The 2-norm of a matrix is equal to the largest singular value:

The 2-norm of the inverse is equal to the reciprocal of the smallest singular value:

Verify using Norm:

The condition number is and thus equals the ratio of largest to smallest singular values:

Properties & Relations (11)

The nonzero singular values of are the square roots of the nonzero eigenvalues of :

Equivalently, they are the square roots of the nonzero eigenvalues of :

The complete set of singular values of is the square root of the eigenvalues of either or :

The product that produces the smaller square matrix, in this case , should be used:

The order that produces a larger square matrix has additional zero eigenvalues:

m and ConjugateTranspose[m] have the same singular values:

The product of the singular values of a square matrix m equals Abs[Det[m]]:

For a normal matrix n, the singular values equal Abs[Eigenvalues[n]]:

MatrixRank[m] equals the number of nonzero singular values:

The singular values of are the reciprocals of the singular values of in the opposite order:

For a submatrix s of m, the largest singular value of m is greater than or equal to the singular values of s:

The generalized singular values of to equal the roots of the generalized eigenvalues of to :

Unlike ordinary singular values, they are not related to the generalized eigenvalues of to :

The largest singular value of with respect to (with having independent columns) is $max_(TemplateBox[{x}, Norm]=1)TemplateBox[{{m, ., x}}, Norm]/TemplateBox[{{a, ., x}}, Norm]$ :