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# SingularValueList

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