If 
, it can be shown that the eigenvalues of C are the
n pairs 
, where 
is a
singular value of A, with 
additional zero eigenvalues if
m > n. The multiplicity of the zero eigenvalue of C is m+n-2r,
where 
.
matrix 
or the 
matrix 
. If
are linearly independent
eigenvectors of 
so that 
, then
is the 
nonzero singular value of A
corresponding to the right singular vector 
. The
corresponding left singular vector, 
, is then
obtained as 
. Similarly, if
are linearly independent
eigenvectors of 
so that 
, then
is the 
nonzero singular value of A
corresponding to the left singular vector 
. The
corresponding right singular vector, 
, is then
obtained as 
.
Computing the SVD(A) using one of the eigenvalue problems
above has its own advantages and disadvantages.
The matrix 
is of order n, whereas the two-cyclic
matrix C defined in Equation (2) is of order 
.
If the matrix A is over-determined, i.e. 
,
the smaller memory requirements for the 
matrix 
make it the more attractive choice for computing the SVD.
However, this scheme only gives the right singular vectors,
and the left singular vectors 
must be obtained by scaling 
.
The eigenvectors of the two-cyclic matrix, on the other hand, are
of the form 
, and
directly give complete
information about the singular triplet 
.
Also, each eigenvalue of 
is 
, forcing a clustering of the singular value
approximations when
. Evaluating the SVD from the eigen-decomposition of
is thus most suited for problems
when only the largest (or dominant) singular values
are desired, with a potential loss
of accuracy for the smaller singular values. Using the
two-cyclic matrix C does not have this drawback, however, at the price of
a larger memory requirement.
