# Calculating frequency and damping ratio from transfer function given eigenvalues

I have the following standard transfer function for a damped linear oscillator:

$$G(s) = \dfrac{\omega_0^2}{s^2 + 2\zeta\omega_0s + \omega_0^2}$$

Now I have two eigen values at locations $-100 \pm 100i$. I want to calculate the damping coefficient, $\zeta$, and natural frequency $\omega_0$.

I thought that the characteristic polynomial of matrix $A$ from the linear system in state space form is the denominator of $G(s)$. However, when solving $s^2 + 2\zeta\omega_0s + \omega_0^2=0$ for $s = -100 - 100i$, I get an equation with two unknowns right? What theoretical point am I missing here? how should I go about this?

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^*)&=s^2-(p+p^*)s+|p|^2\\&=s^2-2\text{Re}\{p\}s+|p|^2\\&=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 $$\zeta=-\cos\phi$$.