Scaling of vectors in non-symmetric generlized eigenproblems

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Scaling of vectors in non-symmetric generlized eigenproblems

Postby andreasnoackjensen » Fri Jul 17, 2015 2:40 pm

I'd like to ask if it is a good idea to scale the vectors of the non-symmetric generalized eigenproblem. The reason is that the scaling destroys the ability to recreate A and B from the eigendecomposition of A-λB. If W are the left vectors and V the right vectors then I think it is convenient and neat that A=(W')^{-1}diag(α)V^{-1} and B=(W')^{-1}diag(β)*V^{-1}. This would also be similar to how the symmetric generalized eigenproblem is normalized.

I've tried to see if I could find a discussion of the scaling in the literature, but so far I haven't succeeded. I presume that the author of xGGEV and xTGEVC preferred to have the values between zero similarly to the usual non-symmetric eigendecomposition, but I think the consequence here is unfortunate. Thank you for your thoughts on the matter.
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