What is known about the statistical independence of the eigenvectors of a real symmetric matrix with independent Gaussian entries with zero mean, and finite variance? The matrix elements are not assumed to have same variance.


I see some results for Wigner matrices in literature, where the entries are i.i.d. standard Gaussian (except diagonal) - though even in this case, whether the eigenvectors are in fact statistically independent is not entirely clear to me (though I suspect that to be the case for Wigner matrices). 


So, does there exist results regarding statistical independence of eigenvectors of random real symmetric matrices with non-identical, but statistically independent Gaussian entries? Any references for this in literature would be helpful.



Statistical independence of eigenvectors of real symmetric Gaussian random matrices
A precise answer exists for the Gaussian Orthogonal Ensemble (all variances the same): then the eigenvectors are the columns of an orthogonal matrix which is uniformly distributed with the Haar measure; they are therefore not independent --- they cannot be because they must be orthogonal to one another.


In the limit $n\rightarrow\infty$ of large $n\times n$ matrices $M$, any finite subset of elements of the eigenvectors does become statistically independent [1] with a Gaussian distribution (mean zero, variance $1/n$). This "central limit" result does not require that the elements of $M$ have identical distributions.


[1] [How Many Entries of a Typical Orthogonal Matrix Can Be Approximated by Independent Normals?](https://www.jstor.org/stable/25449916?seq=1#page_scan_tab_contents) (2006).


