# 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?

• I meant the natural response, considering the basic homogeneous equation – Sagnik Mar 18 at 8:07

\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}