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[Bug-gsl] sign convention for eigenvectors in gsl_eigen_hermv

From: Walter Hahn
Subject: [Bug-gsl] sign convention for eigenvectors in gsl_eigen_hermv
Date: Tue, 18 Feb 2014 18:43:48 +0100
User-agent: Mozilla/5.0 (X11; Linux x86_64; rv:17.0) Gecko/20131103 Icedove/17.0.10

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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