Expectations with self-adjoint random matrix So, we have a square matrix $A=(a_{ij})_{1 \leq i,j \leq n}$ where the entries are independent random variables with the same distribution. Suppose $A = A^{*}$, where $A^{*}$ is the classical adjoint. Moreover, suppose that $E(a_{ij}) = 0$, $E(a_{ij}^{2}) < \infty$. How can I evaluate? $E(Tr A^{2})$?
Clearly, we have $E(Tr A) = 0$ and we can use linearity to get something about $E(Tr A^{2})$ in terms of the entries using simply the formula for $A^{2}$, but for instance I don't see where $A=A^{*}$ comes in... I suppose there's a clever way of handling it...

Clearly
$$
    \operatorname{Tr}(A^2) = \sum_{i,j} a_{i,j} a_{j,i} \stackrel{\rm symmetry}{=}\sum_{i,j} a_{i,j}^2
$$
Thus
$$
   \mathbb{E}\left(\operatorname{Tr}(A^2) \right) = \sum_{i,j} \mathbb{E}(a_{i,j}^2) = n^2 \mathbb{Var}(a_{1,1})
$$
