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