Next: Documentation Design and Program Up: Generalized Eigenvalue and Singular Previous: Generalized Nonsymmetric Eigenproblems (GNEP)   Contents   Index

### 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.
and have the following detailed structures, depending on whether or . In the first case, , then

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.

 Type of Function and storage scheme Real/complex Complex problem Hermitian GSEP simple driver LA_SYGV (real only) LA_HEGV divide and conquer driver LA_SYGVD (real only) LA_HEGVD expert driver LA_SYGVX (real only) LA_HEGVX simple driver (packed storage) LA_SPGV (real only) LA_HPGV divide and conquer driver LA_SPGVD (real only) LA_HPGVD expert driver LA_SPGVX (real only) LA_HPGVX simple driver (band matrices) LA_SBGV (real only) LA_HBGV divide and conquer driver LA_SBGVD (real only) LA_HBGVD expert driver LA_SBGVX (real only) LA_HBGVX GNEP simple driver for Schur factorization LA_GGES expert driver for Schur factorization LA_GGESX simple driver for eigenvalues/vectors LA_GGEV expert driver for eigenvalues/vectors LA_GGEVX GSVD singular values/vectors LA_GGSVD

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 defined2.1.
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].

Next: Documentation Design and Program Up: Generalized Eigenvalue and Singular Previous: Generalized Nonsymmetric Eigenproblems (GNEP)   Contents   Index
Susan Blackford 2001-08-19