I am a graduate student in engineering and we work a lot with so-called Hurwitz (or stable) matrices.

A matrix in our terminology is called stable if the real part of the eigenvalues is strictly negative, see for example here.

Usually the problem for us is that such matrices are not normal, so the spectral theory of these matrices is quite complicated.

I am wondering however about the following property.

Let us assume we can show that there is an approximate eigenvalue, i.e. there is a almost eigenvector $u$ such that $\Vert (A-\lambda I)u \Vert<\varepsilon$ where $\Re (\lambda)=0$ and $\varepsilon>0.$

We engineers say then that $\lambda$ is in the $\varepsilon$ pseudospectrum of $A.$

I would like to know: Does this imply if $A$ is Hurwitz(or stable) that there exists an actual eigenvalue of $A$ in the half-plane $\left\{ \lambda \in \mathbb C: \Re(\lambda) \ge -\varepsilon \right\}$ or does this property above not tell me anything about the location of the spectrum?

I know that this would be true for normal matrices, but of course my matrix is not necessarily normal.