The generalized (or quotient) singular value decomposition of an m-by-n matrix A and a p-by-n matrix B is given by the pair of factorizations
The matrices in these factorizations have the following properties:
and
have the following detailed
structures, depending on whether m - r > = 0 or
m - r < 0. In the first case, m - r > = 0, then
Here l is the rank of B, m = r - 1, C and S are diagonal
matrices satisfying , and S is nonsingular.
We may also identify
,
for
,
, and
for
.
Thus, the first k generalized singular values
are infinite, and the remaining l generalized singular values
are finite.
In the second case, when m - r < 0,
and
Again, l is the rank of B, k = r - 1, C and S are diagonal
matrices satisfying , S 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 B is square and nonsingular, then r = n and the
generalized singular value decomposition of A and B is equivalent
to the singular value decomposition of , where the singular
values of
are equal to the generalized singular values of the
pair A,B:
Second, if
the columns of are orthonormal, then r = n, R = I and the
generalized
singular value decomposition of A and B is equivalent to the CS
(Cosine-Sine) decomposition of
[45]:
Third, the generalized eigenvalues and eigenvectors of
can be expressed in terms of the generalized singular value decomposition:
Let
Then
Therefore, the columns of X 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 n - r columns of X,
which span the common null space of
and
.
The ``trivial eigenvalues'' are not well defined
.
A single driver routine xGGSVD computes the generalized singular value decomposition of A and B (see Table 2.6). The method is based on the method described in [12][10][62].
---------------------------------------------------------------- Type of Function and Single precision Double precision problem storage scheme real complex real complex ---------------------------------------------------------------- GSEP simple driver SSYGV CHEGV DSYGV ZHEGV simple driver SSPGV CHPGV DSPGV ZHPGV (packed storage) simple driver SSBGV CHBGV DSBGV ZHBGV (band matrices) ---------------------------------------------------------------- GNEP simple driver for SGEGS CGEGS DGEGS ZGEGS Schur factorization simple driver for SGEGV CGEGV DGEGV ZGEGV eigenvalues/vectors ---------------------------------------------------------------- GSVD singular values/ SGGSVD CGGSVD DGGSVD ZGGSVD vectors -----------------------------------------------------------------Table 2.6: Driver routines for generalized eigenvalue and singular value problems