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

From: Patrick Alken
Subject: Re: [Bug-gsl] sign convention for eigenvectors in gsl_eigen_hermv
Date: Tue, 18 Feb 2014 10:50:33 -0700
User-agent: Mozilla/5.0 (X11; Linux x86_64; rv:24.0) Gecko/20100101 Thunderbird/24.2.0

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.


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):
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,

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