Let $A$ be an $n\times n$ Hermitian matrix, 
with well-separated eigenvalues $\lambda\_1 > \lambda\_2 ... > \lambda\_n$,
with $|\lambda\_i-\lambda\_j|>\epsilon$, for all $i \neq j$.
Let $G$ be a Gaussian matrix,
i.e. each $G\_{i,j}$ is distributed ${\cal N}(0,1)$.
What can be said about the distribution of the eigenvalues of $A+ f(\epsilon) \cdot G$,
where $f$ is some function $f(\epsilon)<<\epsilon$.
Note that the strength of the perturbation is significantly smaller than the inter-eigenvalue distance, so eigenvalue repulsion can be made arbitrarily weak.


I would like to know whether there are properties that hold for ANY such matrix $A$:


(1) Are the eigenvalues of $A+G$ distributed approximately independently?


(2) Can the variance of the distribution of each eigenvalue of $A+G$ be lower-bounded by some function of $\epsilon$?


If the answer is negative, is there ANY perturbation technique that can yield properties (1) and (2) for any such matrix $A$?



Distribution of the spectrum of a perturbed matrix
Since $P(G)\propto{\rm exp}\bigl(-\frac{1}{2}{\rm Tr}\,GG^{\rm T}\bigr)$ is invariant under orthogonal transformations, if $A$ is real Hermitian you can work in a basis where $A$ is diagonal, with the $\lambda\_n$'s on the diagonal. The first order correction to the $n$-th eigenvalue is then just $\delta\lambda\_{n}= f(\epsilon)G\_{nn}$, and the answer to both of your questions is *Yes*. I would think the same is true if $A$ is complex Hermitian.


