Let $N$ be a self-adjoint matrix and consider a small perturbation of this matrix by another diagonal skew-symmetric matrix $A$ with $0 \in \sigma(A) \subset(-i\infty ,0].$

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

The perturbed matrix $M:=N+A$ is not assumed to have any nice structure besides the fact that we assume that $\Im(\sigma(M)) \le -\delta$ for some $\delta>0$. Here, $\Im$ is the real part and $\sigma$ the spectrum.

Moreover, we assume the following property:

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 $A.$

Also, I am equally interested in making the assumption that that there is an approximate eigenvector $v$ for $N$ such that

$$\Vert (N-\lambda)v \Vert \le \varepsilon \ \text{ and } Av=0.$$

The question:

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

Put differently, can we conclude the existence of an eigenvalue of $M$ close to the real axis?

EDIT: If it helps you may even assume that $A$ is of rank one, i.e. $A=i\mu zz^*.$