Let $N=iT$ be a normal matrix where $T$ is **self adjoint**. 

Now I consider a perturbation of this matrix by another **negative semi-definite self-adjoint matrix** $A.$

If both $N$ and $A$ were self-adjoint, we'd have the nice Weyl inequalities linking the eigenvalues of the sum. 

The perturbed matrix shall be called $M=N+A$ but is not assumed to have any nice structure besides the fact that we assume that $\Re(\sigma(M)) \le -\delta$ for some $\delta>0$.

Moreover, there is a normalized eigenvector $v$ of $N$ with 
$\Vert A v \Vert \le \varepsilon$ for some $\varepsilon,$ i.e. one of the eigenvectors of $N$ is (almost) in the nullspace of $A.$


Does this give us any information about how large $\delta$ can be in terms of $\varepsilon$ or are these two parameters independent?