`gsvd`

Generalized singular value decomposition of a pair of matrices.

Syntax

sigma = gsvd(A, B)

Returns the generalized singular values of A and B.

[U, V, X, C, S] = gsvd(A, B)

Returns the factors with A = U*C*X' and B = V*S*X', where C'*C + S'*S = I.

Description

The GSVD factors two matrices at once with a shared right factor X. It underlies constrained least-squares problems and the comparison of two data sets: a large generalized singular value means a direction in which A dominates B.

It reduces to the ordinary SVD when B is the identity. The ordering of the generalized singular values follows the shape conventions of MATLAB.

Input arguments

A, B: Matrices with the same number of columns.

Output arguments

sigma: Column vector of generalized singular values sqrt(diag(C'*C)) ./ sqrt(diag(S'*S)).
U, V: Unitary matrices.
X: Nonsingular matrix shared by both factorizations.
C, S: Diagonal-like matrices whose squares add up to the identity.

Example

Generalized singular values and a check

A = [1 0; 0 2; 1 1];
B = [1 1; 0 1];
sigma = gsvd(A, B)
[U, V, X, C, S] = gsvd(A, B);
disp(round(C'*C + S'*S))

Output

sigma = 2×1

    1.3028
    2.3028
     1     0
     0     1

gsvd