
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 Σ_{1} [0, R] Q′ and B = V Σ_{2} [0, R] Q′ 
The matrices in these factorizations have the following properties:
In practice, the matrices Σ_{1} and Σ_{2} are never used. Instead a shorter representation with numbers α_{i} and β_{i} is used. These numbers obey 0 ≤ α_{i} ≤ 1 and α_{i}^{2} + β_{i}^{2} = 1. The following relations exist:
Σ_{1}′ Σ_{1} + Σ_{2}′ Σ_{2} = I, 
Σ_{1}′ Σ_{1} = diag (α_{1}^{2}, ..., α_{r}^{2}), and, 
Σ_{2}′ Σ_{2} = diag (β_{1}^{2}, ..., β_{r}^{2}). 
The ratios α_{i} / β_{i} 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