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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 individual eigenvectors distributed (probably zero-mean multi-variate Normal, but what is the covariance)?
  • If $u_i$ and $u_j$ are eigenvectors of $A$, what is the distribution of $|u_i^*u_j|$, or, even better, the $n^2$-d joint distribution $P(u_1, ... u_n)$?
  • What is the joint distribution of eigenvalues and their corresponding eigenvectors (or, perhaps more in line with my application described below, the conditional distribution of an eigenvector given an eigenvalue)?
  • 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 distribution of the number of real eigenvalues of $A$?

Note: I'm not a mathematician, but a physicist working in dynamical systems, and I skipped nuclear so my knowledge of GUE/GOE results is limited to basically the circular laws. I'm really interested in constructing random real matrices $A = VDV^{-1}$ where $D$ is a diagonal matrix of eigenvalues drawn from a distribution that I control and differs from the one given by the various circular laws, and $V$ is the matrix of eigenvectors drawn from the conditional distribution of eigenvectors of random matrices given their corresponding eigenvalue. So this question can be summarized: how do I draw $V$? I don't imagine that there are complete answers to my questions yet, but any insights along those lines that will help me draw $V$ "realistically" would be appreciated. Heck, I just realized bullets two and three may have somewhat incompatible assumptions: bullet three (or, rather, my proposed application) assumes that $P(u_1, ... u_n | \lambda_1, ..., \lambda_n) = P(\lambda_1,...,\lambda_N) \prod_i P(u_i | \lambda_i)$ where $i$ ranges over a single member of each complex conjugate pair of eigenvalues, where two makes no such assumptions and just jumps to $\int P(u_1, ... u_n, \lambda_1, ..., \lambda_n) d\vec{\lambda}$ where $\vec{\lambda}$ is circular law distributed.

If that seems like a weird application, my motivation is to study the influence of only the eigenvalues of the adjacency matrix of a dynamical process that takes place on a random network. A simple first attempt at this by drawing $A$, performing SVD on it to get $V$, and mucking with $D$ only gives either interesting or disastrous results depending on how you look at it. A still simple but second attempt (which to my mind seems like it should work if the conditional independence assumption holds) of drawing several random $A$'s and choosing eigenvalues and corresponding eigenvectors from them according to my desired distribution is even more disastrous (but no more interesting, I think).

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    $\begingroup$ Numerically, I've found that every eigenvector corresponding to a complex eigenvalue has a single real element. -- since you can multiply eigenvectors by an arbitrary scalar constant, this is probably just due to the normalization used in the numerical algorithm and has no real significance. $\endgroup$
    – Federico Poloni
    Commented Oct 31, 2016 at 14:42

3 Answers 3

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If you choose the matrix elements of $A$ independently from a Gaussian distribution you have the socalled Ginibre ensemble of random-matrix theory. The statistics of the eigenvalues is known, see for example Eigenvalue statistics of the real Ginibre ensemble. The statistics of the eigenvectors, and the eigenvector-eigenvalue correlations, have been much less studied, I know of just a few papers:

  1. Eigenvector statistics in non-Hermitian random matrix ensembles

  2. Statistical properties of eigenvectors in non-Hermitian Gaussian random matrix ensembles

  3. Correlations of eigenvectors for non-Hermitian random-matrix models

While in the Hermitian ensembles (GOE, GUE) the eigenvectors of different eigenvalues are independent, in the non-Hermitian ensemble eigenvectors are highly correlated if the two eigenvalues lie close in the complex plane.

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  • $\begingroup$ Cool, thanks. These Ginibre matrices are complex, not real right? Even so, it's a start and I imagine your observation on correlation of eigenvectors probably holds in the real case... I'll leave it unanswered for a few days just in case, though. $\endgroup$
    – Andrew
    Commented Sep 18, 2014 at 8:18
  • $\begingroup$ there are both real and complex Ginibre ensembles; the difference is mainly that the real Ginibre ensemble has a finite fraction $1/\sqrt N$ of eigenvalues on the real axis; away from the real axis the real and complex Ginibre ensembles have the same statistical properties. $\endgroup$
    – Carlo Beenakker
    Commented Sep 18, 2014 at 11:59
  • $\begingroup$ Oops, it was late when I replied and only browsed reference 3 (completely ignoring the title of the inline reference)... answer accepted, thanks! $\endgroup$
    – Andrew
    Commented Sep 18, 2014 at 12:53
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This is just a small update (much later!) on these interesting questions. On the 4th point of the OP, the distribution of the number of real eigenvalues can be found in my pre-print https://arxiv.org/abs/1512.01449.

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Another update: in the paper https://arxiv.org/abs/1608.04923 it is shown how one can use methods from free probability theory to go also beyond the Ginibre ensemble and treat eigenvector correlations for random matrix models corresponding to the single ring theorem

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