John,
certainly, over exact field you don't want to create unnecessary
square roots.
(actually, I would argue against normalisation in fields like QQbar,
as division is expensive there...)
Dima

On Apr 20, 7:42 pm, John Cremona <john.crem...@gmail.com> wrote:
> I would say:  over an inexact field like R or C then it is sensible to
> normalize as Dima suggests (norm 1) rather than making any  one
> nonzero coordinate 1.  But over exact fields (e.g. finite fields,
> number fields) it does make perfect sense to normalise to the first
> (or last?) nonzero coordinate is 1.  And over a field such as Q which
> is the fraction field of a uniqu factorization domain (e.g. Z) one
> coule argue for normalising so that the coordinates were coprime
> integers.
>
> In summary: what is a sensible or desirable normalisation depends a
> lot on what the field is and what sort of mathematics you are doing!
>
> John
>
> On 20 April 2010 12:37, Dima Pasechnik <dimp...@gmail.com> wrote:
>
>
>
> > Dan,
>
> > indeed, it's not too bad to normalize to norm 1, say, but it is quite
> > bad to normalize a given coordinate to 1.
> > I cc this to sage-devel
>
> > Best,
> > Dima
>
> > On Apr 18, 11:21 am, Dan Drake <dr...@kaist.edu> wrote:
> >> On Sat, 17 Apr 2010 at 07:50PM -0700, Dima Pasechnik wrote:
> >> > On Apr 18, 3:29 am, William Stein <wst...@gmail.com> wrote:
> >> > > The test should be rewritten in a way to allow for either sign, since
> >> > > either is correct... or we should re-normalize all the numerical
> >> > > eigenvectors so the first nonzero entry is 1.  I like the idea to
> >> > > normalize.
>
> >> > no, this is a recipe for disaster. If the 1st nonzero entry is close
> >> > to 0, the division by it  will blow the big entries off. Renormalize
> >> > to have the 1st non-0 entry positive, OK. But no division, please,
> >> > unless really necessary.
>
> >> I'm not necessarily arguing for normalization, but I recall that Matlab
> >> normalizes eigenvectors to have length 1. It seems like that kind of
> >> normalization won't behave badly when an entry is very tiny, since the
> >> difference in length is correspondingly tiny.
>
> >> Also, Matlab is very widely used for numerical mathematics, and has been
> >> for a long time. If that kind of normalization was unstable, it's
> >> probably unlikely that it would continue to be used. (Although IANANAE
> >> -- I am not a numerical analysis expert.)
>
> >> (I find it amusing that IANANAE contains "NAN"...)
>
> >> Also, this discussion probably belongs on sage-devel, as it directly
> >> concerns, well, development. :)
>
> >> Dan
>
> >> --
> >> ---  Dan Drake
> >> -----  http://mathsci.kaist.ac.kr/~drake
> >> -------
>
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> >> < 1KViewDownload
>
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