Let $J$ be a $N\times N$ random matrix with a Gaussian measure $$P(J) \propto \exp\big[ -\frac{N}{2(1 - \tau^2)}\mathrm{Tr}(JJ^T - \tau JJ) \big], $$ where $J_{ij}^T = J_{ji}, \mathrm{E}[J_{ij}^2] = \frac{1}{N}, \mathrm{E}[J_{ij}J_{ji}] = \frac{\tau}{N}, \mathrm{E}[J_{ii}^2] = \frac{1 + \tau}{N}$.

A Green's function is defined as follows: $$G(\omega) = \frac{1}{N}\mathrm{E}\big[ \mathrm{Tr}\frac{1}{I\omega - J} \big]$$

It shows 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 $J_{ij}$ in the complex plane.

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

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 ?