Let $A$ be an $n \times n$ random symmetric matrix with $E(A_{i j}) = 0, Var(A_{i j}) = 1/\sqrt n$ for any $i,j$. B is an $n \times n$ symmetric matrix with $B_{ii} = 0$. 

I need to find a upper bound of $Tr(Z^2 (Z\circ H))$ as tight as possible. Notation $\circ$ is Hadamard product and $Tr(\cdot)$ is the trace of a matrix.

Currentely, my approach is as follows. According to [von Neumann's trace inequality](https://en.wikipedia.org/wiki/Trace_inequality#Von_Neumann's_trace_inequality_and_related_results), 

$Tr(Z^2 (Z\circ H)) \le \sum_i \sigma_i(A^2) \sigma_i (A \circ B) = \sum_i \big(\lambda_i(A)\big)^2 \sigma_i (A \circ B)$ , 

where $\sigma_i(\cdot)$ and $\lambda_i(\cdot)$ is the $i$th singular value and eigenvalue of a matrix.

However, I do not know how to handle with $\sigma_i (A \circ B)$. Is there any way to deal with this or is there any better idea to find the upper bound?