- Consider the 2-cyclic matrix
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 . - Alternatively, the SVD(
**A**) can be computed indirectly by the eigenpairs of either the 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.

Michael W. Berry (berry@cs.utk.edu)

Sun May 19 11:34:27 EDT 1996