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 >> ------- >> >> signature.asc >> < 1KViewDownload > > -- > To post to this group, send an email to sage-devel@googlegroups.com > To unsubscribe from this group, send an email to > sage-devel+unsubscr...@googlegroups.com > For more options, visit this group at > http://groups.google.com/group/sage-devel > URL: http://www.sagemath.org > -- To post to this group, send an email to sage-devel@googlegroups.com To unsubscribe from this group, send an email to sage-devel+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/sage-devel URL: http://www.sagemath.org
