[R] SVD/Eigenvector confusion

[Prof Brian Ripley]

On Sun, 29 Feb 2004, Philip Warner wrote:

> At 01:17 AM 29/02/2004, Prof Brian Ripley wrote:
> >On Sun, 29 Feb 2004, Philip Warner wrote:
> >
> > > My understanding of SVD is that, for A an mxn matrix, m > n:
> > >
> > >      A = UWV*
> > >
> > > where W is square root diagonal eigenvalues of A*A extended with zero
> > > valued rows, and U and V are the left & right eigen vectors of A. But this
> > > does not seem to be strictly true and seems to require specific
> > > eigenvectors, and I am not at all sure how these are computed.
> >
> >(A %*% t(A) is required, BTW.)  That is not the definition of the SVD.
> >It is true that U are eigenvectors of A %*% t(A) and V of t(A) %*% A, but
> >that does not make them left/right eigenvectors of A (unless that is your
> >private definition).
> 
> Sorry, that should have read 'left & right singular vectors', and I'm 
> beginning to suspect that they are only the starting point for deriving the 
> singular vectors (based on 
> http://www.cs.utk.edu/~dongarra/etemplates/node191.html)
> 
> 
> >   Since eigenvectors are not unique, it does mean that
> >you cannot reverse the process, as you seem to be trying to do.
> ...cut...
> > >
> > > which seems a little off the mark.
> >
> >It is not expected to work.
> 
> Maybe not by you... 8-}
> 
> 
> 
> >There is no rule: the SVD is computed by a different algorithm.
> 
> So I assume my approach will not give me the singular vectors, and I need a 
> different way of deriving them, is that right?

I think there are ways to derive the correct signs, but your approach is a 
poor way to do the calculations as it squares the condition number of A.

There are standard algorithms for computing the SVD from A alone.

> 
> 
> Thanks for your help, it is much appreciated.
> 
> 
> 
> 
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-- 
Brian D. Ripley,                  ripley at stats.ox.ac.uk
Professor of Applied Statistics,  http://www.stats.ox.ac.uk/~ripley/
University of Oxford,             Tel:  +44 1865 272861 (self)
1 South Parks Road,                     +44 1865 272866 (PA)
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