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 
matrix with 
for 
.
Its entries 
, 
,
are the nonnegative singular
values of 
.
There are several notational conventions in use. A reduced decomposition uses 
and takes 
to be 
, 
to be a 
diagonal matrix,
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 full singular value decomposition has the form
 |
(4)
|
where 
and 
are unitary matrices, 
is the conjugate transpose
of 
, and 
again has nonnegative singular
values in positions 
and zero elsewhere. Such a decomposition
exists for every real matrix or complex
matrix, including rectangular matrices.
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}. Here
U, D, and V are matrices, and the
only possibly nonzero entries of D are the singular values of 
in positions 
.
