generalized singular value decomposition

For m > n, the generalized singular value decomposition (gsvd) of an m × n matrix A and a p × n matrix B is given by the pair of factorizations

A = U D1 [0, R] Q′ and B = V D2 [0, R] Q′

The matrices in these factorizations have the following properties:

• U [m × m], V [p × p] and Q [n × n]
 are orthogonal matrices. In the reconstruction formulas above we maximally need only the first n columns of matrices U and V (when m and/or p are greater than n).
• R [r × r],
is an upper triangular nonsingular matrix. r is the rank of [A′, B′]′ and r ≤ n. The matrix [0, R] is r × n and its first n × (n – r) part is a zero matrix.
• D1 [m × r] and D2 [p × r]
are real, nonnegative and "diagonal".

In practice, the matrices D1 and D2 are never used. Instead a shorter representation with numbers αi and βi is used. These numbers obey 0 ≤ αi ≤ 1 and αi2 + βi2 = 1. The following relations exist:

D1′ D1 + D2′ D2 = I,
D1′ D1 = diag (α12, ..., αr2), and,
D2′ D2 = diag (β12, ..., βr2).

The ratios αi / βi are called the generalized singular values of the pair A, B. Let l be the rank of B and k + l (= r) the rank of [A′, B′]′. Then the first k generalized singular values are infinite and the remaining l are finite. (When B is of full rank then, of course, k = 0).

Special cases

• If B is a square nonsingular matrix, the gsvd of A and B is equivalent to the singular value decomposition of A B–1.

• The generalized eigenvalues and eigenvectors of A′ A - λ B′ B can be expressed in terms of the gsvd. The columns of the matrix X, constructed as

    X = Q*( I 0 )
           ( 0 inv(R) ),

form the eigenvectors. The important eigenvectors, of course, correspond to the positions where the l eigenvalues are not infinite.

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