What is known about the eigenvectors of random matrices?

Question

Let $A$ be a real asymmetric $n \times n$ matrix with i.i.d. random, zero-mean elements. What results, if any, are there for the eigenvectors of $A$? In particular:

• How are the elements of the eigenvectors distributed?
• If $u_i$ and $u_j$ are eigenvectors of $A$, what is the distribution of $|u_i^*u_j|$?
• Numerically, I've found that every eigenvector corresponding to a complex eigenvalue has a single real element. (Naturally, real eigenvalues have corresponding real eigenvectors.) Has this been proven?
• What is the expected number of real eigenvalues of $A$?

(Note: I'm really interested in constructing random matrices $A = VDV^{-1}$ where $D$ is a diagonal matrix of eigenvalues drawn from a distribution that differs from the one given by the various circular laws, and $V$ is the matrix of eigenvectors drawn from the distribution of eigenvectors of random matrices. So this question can be summarized: how do I draw $V$?)

Answer 1

There are some results on this domain: for a general class of i.i.d. symmetric matrices (such as Wigner matrices), it can be shown that with probability one, the matrices would have simple eigenvalues (i.e. $\lambda_1<\lambda_2<\cdots<\lambda_n$). As a result it immediately follows that $u_i^*u_j=0$ for $i\not=j$. Indeed, it is fairly straightforward to show that for classes of matrices the distribution of the eigenvectors is gonna be the Haar measure on the set of orthogonal matrices.

Answer 2

Classic results include the fact the eigenvectors are almost surely delocalised, i.e its almost certainly "spread out" across all the co-ordinate directions. See for instance the works on these by L.Erdos, G.B Arous, etc.