
From:  Walter Hahn 
Subject:  Re: [Buggsl] sign convention for eigenvectors in gsl_eigen_hermv 
Date:  Mon, 10 Mar 2014 12:56:11 +0600 
Useragent:  Mozilla/5.0 (X11; Linux x86_64; rv:24.0) Gecko/20100101 Thunderbird/24.3.0 
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 illposed 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/306717discussion may help. Trying to implement any convention be forcing e.g. a certain sign of the first nonzero element is illdefined 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 bereconsidered. More specifically, I diagonalize a matrix which has only afew nonzero 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 nonzero coefficientsof 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 email 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
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