Talk:Generalized singular value decomposition

It seems to me that Q (in the GSVD of A,B) is not generally unitary, and it looks rather obvious that solution where U,V and Q are all simultaneously unitary does not exist. Unitary matrix on the other hand is orthogonal as far as I understand. This is also mentioned in Alter et al. PNAS 2003 100(6)3351 (http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=152296) where they mention that X^-1 (which corresponds to Q on the wiki page) is 'in general nonorthogonal'. I have not time to check this further right now. However, there seems to be some reason for caution.


 * Hi. Please add a time-stamp for the comments:) As of now, the form of the GSVD is correct, per the paper of Paige and Saunders. I made the definition more accurate today. What I'm wondering though, is whether it would, or would not, make the representation clearer to further decompose $$X = W^* R$$, where $$W$$ is unitary and $$R$$ is upper-triangular. This is how it reads in Paige and Saunders. However, that form can always be obtained by the QR-decomposition of $$X$$, so I find it clearer not to introduce such additional matrices. On the other hand, LAPACK documentation also uses the upper-triangular form. --Kaba3 (talk) 10:00, 7 February 2013 (UTC)