[Numpy-discussion] One question about the numpy.linalg.eig() routine

[Val Kalatsky]

Interesting. I happen to know a little bit about Berry's phase
http://keck.ucsf.edu/~kalatsky/publications/PRL1998_BerryPhaseForLargeSpins.pdf
http://keck.ucsf.edu/~kalatsky/publications/PRA1999_SpectraOfLargeSpins-General.pdf
The latter one knocks out all point groups.
Probably you want to do something different, I cared about eigenvalues only
(BTW my Hamiltonians were carefully crafted).
Cheers
Val

PS I doubt anybody on this list cares to hear more about Berry's phase,
should take this discussion off-line


2012/4/3 Hongbin Zhang <hongbin_zhang82 at hotmail.com>

>  Hej Val,
>
> Thank you very much for your replies.
>
> Yes, I know that both eigenvectors are correct while they are indeed
> related
> to each other by unitary transformations (unitary matrices).
>
> Actually, what I am trying to do is to evaluate the Berry phase which is
> closely related to the gauge chosen. It is okay to apply an arbitrary
> phase to the eigenvectors, while to get the (meaningful) physical quantity
> the phase should be consistent for all the other eigenvectors.
>
> To my understanding, if I run both Fortran and python on the same computer,
> they should have the same phase (that is the arbitrary phase is
> computer-dependent). Maybe some additional "rotations" have been performed
> in
> python,
> but should this be written/commented somewhere in the man page?
>
> I will try to fix this by performing additional rotation to make the
> diagonal
> elements real and check whether this is the solution or not.
>
> Thank you all again, a nd of course more insightful suggestions are
> welcome.
>
> Regards,
>
>
> Hongbin
>
>
>
>                                                         Ad hoc, ad loc
> and quid pro quo
>
>          &n bsp;
>                                           ---   Jeremy Hilary Boob
>
>
> ------------------------------
> Date: Mon, 2 Apr 2012 22:19:55 -0500
> From: kalatsky at gmail.com
> To: numpy-discussion at scipy.org
> Subject: Re: [Numpy-discussion] One question about the numpy.linalg.eig()
> routine
>
>
> BTW this extra degree of freedom can be used to "rotate" the eigenvectors
> along the unit circle (multiplication by exp(j*phi)). To those of physical
> inclinations
> it should remind of gauge fixing (vector potential in EM/QM).
> These "rotations" can be used to make one (any) non-zero component of each
> eigenvector be positive real number.
> Finally to the point: it seems that numpy.linalg.eig uses these
> "rotations" to turn the
> diagonal elements in the eigenvector matrix to real positive numbers,
> that's why the numpy solutions looks neat.
> Val
>
> PS Probably nobody cares to know, but the phase factor I gave in my 1st
> email should be negated:
> 0.99887305445887753+0.047461785427773337j
>
> On Mon, Apr 2, 2012 at 8:53 PM, Matthew Brett <matthew.brett at gmail.com>wrote:
>
> Hi,
>
> On Mon, Apr 2, 2012 at 5:38 PM, Val Kalatsky <kalatsky at gmail.com> wrote:
> > Both results are correct.
> > There are 2 factors that make the results look different:
> > 1) The order: the 2nd eigenvector of the numpy solution corresponds to
> the
> > 1st eigenvector of your solution,
> > note that the vectors are written in columns.
> > 2) The phase: an eigenvector can be multiplied by an arbitrary phase
> factor
> > with absolute value = 1.
> > As you can see this factor is -1 for the 2nd eigenvector
> > and -0.99887305445887753-0.047461785427773337j for the other one.
>
> Thanks for this answer; for my own benefit:
>
> Definition: A . v = L . v  where A is the input matrix, L is an
> eigenvalue of A and v is an eigenvector of A.
>
> http://en.wikipedia.org/wiki/Eigendecomposition_of_a_matrix
>
> In [63]: A = [[0.6+0.0j,
> -1.97537668-0.09386068j],[-1.97537668+0.09386068j, -0.6+0.0j]]
>
> In [64]: L, v = np.linalg.eig(A)
>
> In [66]: np.allclose(np.dot(A, v), L * v)
> Out[66]: True
>
> Best,
>
> Matthew
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