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 Σ1 [0, R] Q′ and B = V Σ2 [0, R] Q′|
The matrices in these factorizations have the following properties:
In practice, the matrices Σ1 and Σ2 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:
|Σ1′ Σ1 + Σ2′ Σ2 = I,|
|Σ1′ Σ1 = diag (α12, ..., αr2), and,|
|Σ2′ Σ2 = 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).
• 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.
© djmw, October 7, 1998