# What is known about the eigenvectors of random matrices?

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 $V$?)

• Every eigenvector has one real element because you are free to multiply a normalised eigenvector by an overall phase, and thus are free to set one element to be real. There is no meaning to it, it is just an artefact of choice made by the numerical solver. Oct 2, 2021 at 2:12

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.