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Suppose a random matrix $A$ with its elements following Gaussian distribution with non-zero mean. We know that the eigenvalues of $A$ have two patches: one is at the real axis that is far away from the origin point, and the other one is in a circle around the origin point. We know that the eigenvector of the first patch is $[1,1,\cdots,1]$, but what is the distribution of the elements of an eigenvector whose corresponding eigenvalue is in the second patch?

Here is a picture of the situation:

An image.

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  • $\begingroup$ This question makes no sense to me. $\endgroup$
    – Igor Rivin
    Commented Nov 19, 2015 at 14:47
  • $\begingroup$ The picture is misleading. It shows the locations of eigenvalues, but the question is about something about the eigenvectors. The $\{1,1,...,1\}$ is an approximation. Are you normalizing the eigenvectors, and then do you want to know the distribution of each coordinate? $\endgroup$
    – Douglas Zare
    Commented Nov 20, 2015 at 12:55
  • $\begingroup$ @Douglas Yes, that is what I mean. $\endgroup$
    – Zedong Bi
    Commented Nov 21, 2015 at 8:42
  • $\begingroup$ I would be surprised if the marginal distribution of a few of the elements of an eigenvector would be much different from independent Gaussians; have you tried generating some random matrices and testing for this? $\endgroup$
    – Carlo Beenakker
    Commented Nov 21, 2015 at 9:24
  • $\begingroup$ The distribution shown in the picture is the result of a test. We see that it is indeed similar to Gaussian. $\endgroup$
    – Zedong Bi
    Commented Nov 23, 2015 at 1:54

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