What is the relation between eigenvalues and state-space response in control systems? I understand the mathematics behind it but I want to know what happens physically in a real-life system. How do the eigenvalues come into the picture from a non-mathematical (physical) point of view? In linear algebra, a matrix is basically a linear transformation and eigenvectors are those which, on being applied this transformation, just stretches themselves (that is without any rotation, I'm limiting only to 2D space). So when I consider a physical system (I'm only considering a homogeneous or unforced system to make it simpler) what do these eigenvalues or eigenvectors physically represent and how do they relate to their mathematical interpretations?
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.