Distribution of the spectrum of a perturbed matrix

Question

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 $A$?

Answer 1

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.