If a matrix 
has a matrix of eigenvectors 
that is not invertible (for
example, the matrix ![[1 1; 0 1]](/images/equations/SingularValueDecomposition/Inline3.svg)
has the noninvertible system of eigenvectors ![[1 0; 0 0]](/images/equations/SingularValueDecomposition/Inline4.svg)
),
then 
does not have an eigen decomposition. However,
if 
is an 
real matrix with 
, then 
can be written using a so-called singular value decomposition
of the form
 |
(1)
|
Note that there are several conflicting notational conventions in use in the literature. Press et al. (1992) define 
to be an 
matrix, 
as 
, and 
as 
. However, the Wolfram
Language defines 
as an 
, 
as 
, and 
as 
. In both systems, 
and 
have orthogonal columns
so that
 |
(2)
|
and
 |
(3)
|
(where the two identity matrices may have different dimensions), and 
has entries only along the diagonal.
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 
is a diagonal matrix whose elements are the singular values of the original matrix. If 
is a complex matrix, then
there always exists such a decomposition with positive singular values (Golub and
Van Loan 1996, pp. 70 and 73).
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 
.
