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 Σ1 [0, R] Q′ and B = V Σ2 [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 formula's 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 rn. The matrix [0, R] is r × n and its first n × (nr) part is a zero matrix.
Σ1 [m × r] and Σ2 [p × r]
are real, nonnegative and "diagonal".

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:

Σ1Σ1 + Σ2Σ2 = I,
Σ1Σ1 = diag (α12, ..., αr2), and,
Σ2Σ2 = 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 AA - λ BB 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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© djmw, October 7, 1998