Link between generalized singular value decomposition (GSVD) and generalized eigenvalue problem.

Hi everyone,

I meet with some difficulties when I try to point out the link between the generalized singular value decomposition, GSVD, and the generalized eigenvalue problem.

The former method applied on two data matrices, let say A and B, returns through gsvd(A,B):

A = U*C*X'
B = V*S*X'
C'*C + S'*S = I

So, we have,

AA’ = XC^2X’,
B’B = XS^2X’,

meaning that the columns of my X matrix are the eigenvectors of the covariance matrices of A and B.

On the other hand, if the same matrices are used, the generalized eigenvalue problem can be solved by eig(A*A’,B*B’) and is formulated as

A*A’ v = \lambda B*B’ v

Is there any link between the both?

Answer 1

[Christopher Judge]

See section 8.7.3 of the text "Matrix Computations" by Golub and Van Loan. C and S are diagonal matrices with entries and . The square of the ratio of each pair of diagonal elements, , is an eigenvalue of the problem , and the columns of X are the eigenvectors of this generalized eigenvalue problem.

So these problems are not only linked. They are essentially the same problem!

According to Golub and Van Loan:

"The value of the GSVD is that these diagonalizations can be achieved without forming and .