What is the relation between eigenvalues and state-space response in control systems?
Just because eigenvalues apply in linear algebra doesn't mean that the same meaning applies everywhere. Think of the Fourier transform: does it only apply to the study of heat transfer? Same here. The A matrix is the companion matrix, which means its eigenvalues represent the roots of the system. For example:
$$\begin{align}
H(s)&amp;=\dfrac{3}{s^2+2s+3}\tag{1} \\
\mathbf{A}&amp;=\begin{bmatrix}{{0}\quad{-3} \\ {1}\quad{-2}}\end{bmatrix}\tag{2} \\
s_{1,2}&amp;=-1\pm j\sqrt{2}\tag{3} \\
\end{align}$$
I'm afraid I don't quite understand what you mean by &quot;from a non-mathematical (physical) point of view&quot;. The eigenvalues are the mathematical roots of the system, they tell about the stability, the time and frequency response. There's nothing non-physical about it. The system itself is physical -- as in it has a meaning -- even if only described on paper, such as the example above. It's just that you don't have to think about them from a single perspective, only.