The singular value decomposition of an 
real matrix 
is a factorization
 |
(1)
|
In the full decomposition, 
is an 
orthogonal matrix,
is an 
orthogonal matrix, and 
is an 
diagonal matrix
whose diagonal entries are the nonnegative singular
values of 
.
There are several notational conventions in use. A reduced decomposition uses 
and takes 
to be 
,
to be 
,
and 
to be 
.
In either convention, the columns retained in 
and 
are orthonormal, so
that
 |
(2)
|
and
 |
(3)
|
for the appropriate identity matrices (Golub and Van Loan 1996, pp. 70-71).
For a complex matrix 
, the singular value decomposition is a decomposition into
the form
 |
(4)
|
where 
and 
are unitary matrices, 
is the conjugate transpose
of 
,
and 
again has nonnegative singular values on its diagonal. Such a decomposition exists
for every real or complex rectangular matrix.
When 
is the coefficient matrix of a vector
in a tensor product of two Hilbert
spaces, its singular value decomposition gives the Schmidt
decomposition of that vector.
Singular value decomposition is implemented in the Wolfram Language as SingularValueDecomposition[m],
which returns a list 
U, D, V
, where U and V are matrices and D is
a diagonal matrix made up of the singular values of 
.
