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.](https://i.sstatic.net/hCGhv.jpg)