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Generalized Singular Value Decomposition (GSVD)

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:

- U is m-by-m, V is p-by-p, Q is n-by-n, and all three matrices are orthogonal. If A and B are complex, these matrices are unitary instead of orthogonal, and QT should be replaced by QH in the pair of factorizations.
- R is r-by-r, upper triangular and nonsingular. [0,R] is r-by-n (in other words, the 0 is an r-by-n-r zero matrix). The integer r is the rank of , and satisfies .
- is m-by-r, is p-by-r, 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 m-r < 0. In the first case, , then

Here l is the rank of B, k=r-l, C and S are diagonal matrices satisfying C2 + S2 = 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 m-r < 0,

and

Again, l is the rank of B, k=r-l, C and S are diagonal matrices satisfying C2 + S2 = 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 (Cosine-Sine) 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 n-r columns of X, which span the common null space of AT A and BT B. The ``trivial eigenvalues'' are not well defined2.1.

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