A Green's function is defined as follows: $$G(\omega) = \frac{1}{N}\mathrm{E}\big[ \mathrm{Tr}\frac{1}{I\omega - J} \big]$$, where $I$ is the $N$-dimensional identity and $E$ means expectation value with respect to the random matrix $J$.

It follows that $$ G(\omega) = \frac{1}{N}\mathrm{E}\big[ \sum_\lambda\frac{1}{\omega - \lambda} \big] = \int d^2\lambda\frac{\rho(\lambda)}{\omega - \lambda},$$ where $\rho(\lambda)$ is the average density of eigenvalues $\lambda$ of $J$ in the complex plane.

I am confused as to how this equality is obtained. Can someone help to explain or present the idea behind it ?