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 should be replaced by 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 h-by-n - r zero matrix). The integer r is the rank of , and satisfies r < = n.
• 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 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].

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Type of  Function and         Single precision  Double precision
problem  storage scheme       real     complex  real     complex
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GSEP     simple driver        SSYGV    CHEGV    DSYGV    ZHEGV

         simple driver        SSPGV    CHPGV    DSPGV    ZHPGV
          (packed storage)

         simple driver        SSBGV    CHBGV    DSBGV    ZHBGV
          (band matrices) 
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GNEP     simple driver for    SGEGS    CGEGS    DGEGS    ZGEGS
          Schur factorization 

         simple driver for    SGEGV    CGEGV    DGEGV    ZGEGV
          eigenvalues/vectors 
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GSVD     singular values/     SGGSVD   CGGSVD   DGGSVD  ZGGSVD
         vectors 
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