Related Singular Value Problems

- If , and the SVD of
, then the SVD of
.
- Suppose , where is by with .
Then is an by Hermitian matrix.
Let
be the SVD of .
Then the eigendecomposition of
.
Note that
.
In other words, the eigenvectors of are the right singular vectors of ,
and the eigenvalues of are the squares of the singular values of .
If , where is by with , then is by
with eigendecomposition
. In other words,
the eigenvectors of are the right singular vectors of ,
and the eigenvalues of are the squares of the singular values of ,
in addition to 0 with additional multiplicity .
- Suppose
,
where is by with .
Then is an by Hermitian matrix.
Let
be the SVD of .
Write
and
.
Then the eigendecomposition of
, where
and

In other words, is an eigenvalue with unit eigenvector for to , and 0 is an eigenvalue with eigenvector for . - The
*QSVD or generalized SVD (GSVD)*of and is defined as follows. Suppose is by and is by and nonsingular. Let . Let the SVD of . We may also write two equivalent decompositions of and as and , where is by and nonsingular, is by and contains in its leading rows and columns and zeros elsewhere, and is by and contains .

and can be chosen so that . Thus . The diagonal entries of are called the*generalized singular values*of and .This decomposition generalizes to the cases where is singular or by , to a decomposition equivalent to the SVD of (the

*product SVD*), and indeed to decompositions of arbitrary products of the form [108]. - The
*CS (cosine/sine) decomposition*of and is defined as follows. Suppose the columns of are orthonormal. The QSVD of and is then equivalent to the decompositions and , where is unitary. The diagonal entries of () are the cosines (sines) of the*principal angles*between the subspace spanned by the columns of and the subspace spanned by the first columns of [424,25]. This is used to compute the ``distance'' (or angles) between subspaces. - Suppose is -by-, and is by with full column rank;
then the GHEP
can be solved using the QSVD as follows. Write the QSVD
as
and
. The eigendecomposition
of
may be written
and
. This generalizes to the case of
without full column rank.
- Finding that minimizes
is the
*linear least squares problem.*Suppose is by with ; we say that the least squares problem is*overdetermined*. Let be the compact SVD of . If has full rank, the unique solution of the least squares problem is , although solutions based on the QR decomposition or normal equations are cheaper and more commonly used [50,12]. If 's rank is less than , then the SVD is often used to solve the least squares problem. In this case the solution is not unique but its unique*minimum norm*solution is

If we say that the least squares problem is*underdetermined*. In this case the solution is not unique but its unique*minimum norm*solution is , where is the compact SVD of .