I want to compute the expected value of $$ E[A^TBA] $$ where $A$ is a random variable (matrix) from Normal distribution: $$ A=\begin{bmatrix} a_{11} & a_{12} & a_{13} & \dots & a_{1N} \\ a_{21} & a_{22} & a_{23} & \dots & a_{2N} \\ &&...&& \\ a_{M1} & a_{M2} & a_{M3} & \dots & a_{MN} \end{bmatrix} $$ $$ a_{ij}\sim N(\mu_A,\sigma_A) $$ and $B$ is a random variable (matrix $M*M$) from Wishart distribution: $$ B\sim W(w,\Sigma_B) $$ Also, $A$ and $B$ are independent and as you may know: $$E[B]=w\Sigma_B$$
would you please help me in solving $E[A^TBA]$?