Random complex eigenvalues and averages of traces

Question

Suppose I have a random matrix $M$ which is real, but not symmetric. Suppose I know that the marginal distribution of its eigenvalues is uniform over the unit disk in the complex plane. What does that tell me about the average values of traces, $\langle {\rm Tr}(M^n)\rangle$, $\langle {\rm Tr}(M^n){\rm Tr}(M^m)\rangle$, etc?

Answer 1

$$\newcommand{\Tr}{\mathrm{Tr}}$$ Here's a hint: Enumerate the eigenvalues of $M$ as $\lambda_1,\ldots,\lambda_k$ (note that these are random). Then $$\Tr(M^n) = \lambda_1^n + \cdots + \lambda_k^n\,.$$ Expand this and take expectations.