# The distribution of the elements of an eigenvector of random matrices

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:

• This question makes no sense to me. Nov 19, 2015 at 14:47
• 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? Nov 20, 2015 at 12:55
• @Douglas Yes, that is what I mean. Nov 21, 2015 at 8:42
• 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? Nov 21, 2015 at 9:24
• The distribution shown in the picture is the result of a test. We see that it is indeed similar to Gaussian. Nov 23, 2015 at 1:54