What is the relation between eigenvalues and state-space response in control systems?

Question

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?

Answer 1

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:

$$\begin{align} H(s)&=\dfrac{3}{s^2+2s+3}\tag{1} \\ \mathbf{A}&=\begin{bmatrix}{{0}\quad{-3} \\ {1}\quad{-2}}\end{bmatrix}\tag{2} \\ s_{1,2}&=-1\pm j\sqrt{2}\tag{3} \\ \end{align}$$

I'm afraid I don't quite understand what you mean by "from a non-mathematical (physical) point of view". 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.