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Generalized Singular Value Decomposition (GSVD)
The generalized (or quotient) singular value decomposition
of an mbyn matrix A and a
pbyn matrix B is given by the pair of factorizations
The matrices in these factorizations have the following properties:
 U is mbym, V is pbyp, Q is nbyn, and
all three matrices are orthogonal. If A and
B are complex, these matrices are unitary instead of
orthogonal, and Q^{T} should be
replaced by Q^{H} in the pair of factorizations.
 R is rbyr, upper triangular and nonsingular.
[0,R] is rbyn (in other words, the 0 is an rbynr
zero matrix).
The integer r is the rank of
,
and satisfies .

is mbyr,
is pbyr, 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 A, B.
If ,
then the generalized singular value
is infinite.
and
have the following detailed
structures, depending on whether
or
mr < 0. In the first case, ,
then
Here l is the rank of B, k=rl, C and S are diagonal
matrices satisfying
C^{2} + S^{2} = I, 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 mr < 0,
and
Again, l is the rank of B, k=rl, C and S are diagonal
matrices satisfying C^{2} + S^{2} = I, S is nonsingular,
and we may identify
,
for
,
,
,
for
,
and
.
Thus, the first k generalized singular values
are infinite, and the remaining l 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 AB^{1}, where the singular
values of AB^{1} 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
(CosineSine) decomposition of
[55]:
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 2.3.5.1).
``Trivial'' eigenvalues
are those corresponding to the leading nr columns of X,
which span the common null space of A^{T} A and B^{T} B.
The ``trivial eigenvalues'' are not well defined^{2.1}.
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
[83,10,8].
Table 2.6:
Driver routines for generalized eigenvalue and singular value problems
Type of 
Function and storage scheme 
Single precision 
Double precision 
problem 

real 
complex 
real 
complex 
GSEP 
simple driver 
SSYGV 
CHEGV 
DSYGV 
ZHEGV


divide and conquer driver 
SSYGVD 
CHEGVD 
DSYGVD 
ZHEGVD


expert driver 
SSYGVX 
CHEGVX 
DSYGVX 
ZHEGVX


simple driver (packed storage) 
SSPGV 
CHPGV 
DSPGV 
ZHPGV


divide and conquer driver 
SSPGVD 
CHPGVD 
DSPGVD 
ZHPGVD


expert driver 
SSPGVX 
CHPGVX 
DSPGVX 
ZHPGVX


simple driver (band matrices) 
SSBGV 
CHBGV 
DSBGV 
ZHBGV


divide and conquer driver 
SSBGVD 
CHBGVD 
DSBGV 
ZHBGVD


expert driver 
SSBGVX 
CHBGVX 
DSBGVX 
ZHBGVX

GNEP 
simple driver for Schur factorization 
SGGES 
CGGES 
DGGES 
ZGGES


expert driver for Schur factorization 
SGGESX 
CGGESX 
DGGESX 
ZGGESX


simple driver for eigenvalues/vectors 
SGGEV 
CGGEV 
DGGEV 
ZGGEV


expert driver for eigenvalues/vectors 
SGGEVX 
CGGEVX 
DGGEVX 
ZGGEVX

GSVD 
singular values/vectors 
SGGSVD 
CGGSVD 
DGGSVD 
ZGGSVD

Next: Computational Routines
Up: Generalized Eigenvalue and Singular
Previous: Generalized Nonsymmetric Eigenproblems (GNEP)
Contents
Index
Susan Blackford
19991001