Stable matrices and their spectra

Question

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.

Answer 1

No, this is not true in general.

Note that in your condition you probably want to assume $\|u\|=1$ for otherwise you could always make the left hand side as small as you wish by making $u$ small. I will answer the question for the Euclidean norm and the induced spectral norm but this does not really make a huge difference.

Aassuming $\|u\|=1$, all that we learn from the condition is that $(A-\lambda I)u = y$ with $\|y\| < \varepsilon$. This can be rewritten to say $$ (A - yu^T)u = \lambda u$$ so that a perturbation of $A$ of size less than $\varepsilon$ moves one eigenvalue to the imaginary axis. Note that $\| yu^T\|=\|y\|$.

Now as you suspect, for nonnormal matrices perturbations of size $\varepsilon$ can have significant effects on the spectrum and there is no reason to assume that the effect is itself bounded by $\varepsilon$. To get more information you need to know more about $A$ and possibly about the type of perturbations that are of interest. There is a wealth of literature on this. As you are from engineering I suggest to read up on the concepts stability radii, spectral value sets and some of the introductory papers by Trefethen.

A small comment aside: you say that "we engineers say $\lambda$ is in the $\varepsilon$-pseudospectrum of $A$"; historically the term was first coined by mathematicians.