Re: [Bug-gsl] sign convention for eigenvectors in gsl_eigen_hermv

[Walter Hahn]

Dear Håkan and Patrick,

thank you very much for your responses. I agree with you on all you have 
written about eigenvectors but I nevertheless expected that there should 
be a way to establish a sign convention. In fact, I meant it actually in 
terms of consistency, i.e., that if a matrix has identical blocks on the 
diagonal, the corresponding eigenvectors will also be identical.

However, I have realised that there is no need to introduce a sign 
convention (if possible at all), even in terms of consistency, and that 
my problem was actually an ill-posed one.

Thank you,
Walter



Am 18.02.2014 23:54, schrieb Håkan Johansson:
> Perhaps also the answer dated 23 Apr, 2011 09:28:04 in the following
>
> http://www.mathworks.com/matlabcentral/newsreader/view_thread/306717
>
> discussion may help.  Trying to implement any convention be forcing 
> e.g. a certain sign of the first non-zero element is ill-defined and 
> therefore hazardous.
>
> Cheers,
> Håkan
>
>
> On Tue, 18 Feb 2014, Patrick Alken wrote:
>
> > In case you didn't see my reply to your first message, it is pasted 
> > below:
> >
> > -----
> >  There is no "sign convention" regarding eigenvectors, since if
> > v is an eigenvector of a matrix, any scalar multiple of v is also an
> > eigenvector. gsl_eigen_hermv guarantees that the eigenvectors are
> > normalized to unit magnitude, but of course the negative of the computed
> > vector is still a valid eigenvector with unit magnitude.
> >
> > I suspect you will find that any eigenvector software will exhibit the
> > same behavior. There really is no way to determine a standard sign
> > convention for eigenvectors.
> > -----
> >
> > Your vector v8 is a perfectly legitimate eigenvector of the matrix.
> >
> > Patrick
> >
> > On 02/18/2014 10:43 AM, Walter Hahn wrote:
> > > Dear all,
> > >
> > > a few weeks ago, I have sent you this bug report already without giving
> > > you enough information. Now, I would like to provide the missing
> > > information and correct some statements.
> > >
> > > After using the procedure gsl_eigen_hermv to diagonalize a matrix, I
> > > suspect that the sign convention for the eigenvectors should be
> > > reconsidered. More specifically, I diagonalize a matrix which has 
> > > only a
> > > few non-zero entries, namely at m(2i,2i+1) and m(2i+1,2i) for all i. In
> > > my case, I diagonalize a 8x8 matrix with real entries, e.g. 1.0.
> > >
> > > The eigenvectors should be of the following form (not normalized):
> > > v1=(1,1,0,0,0....)
> > > v2=(-1,1,0,0,0,...),
> > > other eigenvectors can be obtained by shifting the non-zero 
> > > coefficients
> > > of v1 and v2 by two places to the right.
> > >
> > > However, diagonalizing the 8x8 matrix described above, I obtain the
> > > eigenvectors as described above except the last one which is
> > > v8=(0,0,0,0,0,0,1,-1) instead of (0,0,0,0,0,0,-1,1), i.e., multiplied
> > > with (-1). Therefore, I think that the sign convention is either not
> > > implemented correctly or the convention used is not broad enough.
> > >
> > > Please find in the attachment to this e-mail a simple compilable code
> > > which demonstrates this problem. I have checked the described results
> > > for GSL versions 1.15 and 1.16.
> > >
> > > Kind regards,
> > > Walter