Generalized Singular Value Decomposition (GSVD)

The **generalized (or quotient) singular value decomposition**
of an matrix and a
matrix is given by the pair of factorizations

The matrices in these factorizations have the following properties:

- is , V is , is , and
all three matrices are orthogonal. If and
are complex, these matrices are unitary instead of
orthogonal, and should be
replaced by in the pair of factorizations.

- is , upper triangular and nonsingular.
is (in other words, the is an
zero matrix).
The integer is the rank of
, and satisfies .

- is ,
is , both are real, nonnegative and diagonal, and
.
Write
and
,
where and lie in the interval from 0 to 1.
The ratios
are called the
**generalized singular values**of the pair , . If , then the generalized singular value is**infinite**.

Here is the rank of , , and are diagonal matrices satisfying , and is nonsingular. We may also identify , for , , and for . Thus, the first generalized singular values are infinite, and the remaining generalized singular values are finite.

In the second case, when ,

and

Again, is the rank of , , and are diagonal matrices satisfying , is nonsingular, and we may identify , for , , , for , and . Thus, the first generalized singular values are infinite, and the remaining generalized singular values are finite.

Here are some important special cases of the generalized singular value decomposition. First, if is square and nonsingular, then and the generalized singular value decomposition of and is equivalent to the singular value decomposition of , where the singular values of are equal to the generalized singular values of the pair , :

Second, if the columns of are orthonormal, then , and the generalized singular value decomposition of and is equivalent to the CS (Cosine-Sine) decomposition of [20]:

Third, the generalized eigenvalues and eigenvectors of can be expressed in terms of the generalized singular value decomposition: Let

Then

Therefore, the columns of are the eigenvectors of , and the ``nontrivial'' eigenvalues are the squares of the generalized singular values (see also section 2.2.5.1). ``Trivial'' eigenvalues are those corresponding to the leading columns of , which span the common null space of and . The ``trivial eigenvalues'' are not well defined

A single driver routine LA_GGSVD computes the generalized singular value decomposition of and (see Table 2.6). The method is based on the method described in [33,2,3].