Calculating frequency and damping ratio from transfer function given eigenvalues
If I understand you correctly your system has two complex conjugate poles at $p=-100+100i$ and $p^*=-100-100i$ (where $^*$ denotes complex conjugation). Consequently, your denominator polynomial can be written as
$$\begin{align}(s-p)(s-p^*)&amp;=s^2-(p+p^*)s+|p|^2\\&amp;=s^2-2\text{Re}\{p\}s+|p|^2\\&amp;=s^2-2|p|\cos(\phi)s+|p|^2\end{align}\tag{1}$$
where $\phi$ is the pole angle: $p=|p|e^{i\phi}$. For the given poles you have $\phi=3\pi/4$. Comparing $(1)$ with the denominator in your question immediately gives $\omega_0=|p|$ and $\zeta=-\cos\phi$.