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...

Identical as Sasha's reply: via Hilbert-Schmidt norms,
$$\operatorname{Tr}(A^2) = \operatorname{Tr}(A^\top A) = \|A\|_{\text HS}^2 = \sum_{i,j} a_{i,j}^2$$
