BUILT-IN MATHEMATICA SYMBOL

SingularValueList

SingularValueList[m]
gives a list of the nonzero singular values of a matrix m.

SingularValueList

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

SingularValueList

gives the k largest singular values of m.

SingularValueList

gives the k largest generalized singular values of m.

MORE INFORMATION

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 , 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[Conjugate[Transpose[m]].m]].

EXAMPLES

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Basic Examples (1)

There are only two nonzero singular values since the matrix is singular:

In[1]:=

Out[1]=

Scope (3)

Exact input:

m is a 4×3 matrix:

Singular values computed using exact arithmetic:

Singular values computed using machine arithmetic:

Singular values computed using 30-digit precision arithmetic:

Singular values for a random complex-valued 2×5 matrix:

Generalizations & Extensions (2)

Find the largest 3 singular values for a large sparse matrix:

Symbolic input:

Options (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 (1)

m is a random 10×10 matrix:

Find the singular values of m:

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:

The condition number of the matrix is equal to the ratio of largest to smallest singular values:

Properties & Relations (1)

m is a matrix of random size having random entries:

Find the singular values of m:

These are equal to the square roots of the nonzero eigenvalues of m.Transpose[m]:

Possible Issues (1)

s is a large sparse matrix:

Computing all of the singular values uses dense linear algebra, which may be prohibitive:

Computing just a few will typically be much faster: