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SingularValues
SingularValues[
m
] gives the singular value decomposition for a numerical matrix m. The result is a list 
u
,
w
,
v
 , where w is the list of singular values, and m can be written as Conjugate[Transpose[
u
]].DiagonalMatrix[
w
].
v.
SingularValues[
m
,
Tolerance
->
t
] specifies that singular values smaller than t times the maximum singular value are to be removed.
The default setting Tolerance
->
Automatic typically takes t to be  where  is the numerical precision of the input.
With Tolerance->0 singular values which are exactly zero can be returned.
u and v are row orthonormal matrices, which can be considered as lists of orthonormal vectors.
The ratio of the largest to smallest singular value gives the  condition number of m.
See the Mathematica book: Section 3.7.10.
See also Implementation NotesA.9.44.27MainBookLinkOldButtonDataA.9.44.27.
See also: PseudoInverse, QRDecomposition, SchurDecomposition, LUDecomposition.
Related packages: LinearAlgebra`Cholesky`, Statistics`LinearRegression`.
Further Examples
This is the singular value decomposition of a non-singular  x  inexact matrix.
In[1]:= 
Out[1]= 
This checks the result.
In[2]:= 
Out[2]//MatrixForm= 
This is the singular value decomposition of a singular matrix. Only one singular value is given.
In[3]:= 
Out[3]= 
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