Suppose we have $$ x\sim Normal(x|0,\Sigma) $$ where $x\in R^N$ and $$ \Sigma\sim Wishart(\Sigma|w,\Psi) $$ where $w\in R$ and $\Psi$ is a $N\times N$ semi-positive matrix.
How can we compute $E[x^T\Sigma x]$?
the solution may be started as below
$$ x^T\Sigma x=\begin{bmatrix} x_1,...,x_N \end{bmatrix} \begin{bmatrix} \sigma_{11}^2 & \sigma_{12}^2 & \dots & \sigma_{1N}^2 \\ \sigma_{21} ^2 & \sigma_{22}^2 & \dots & \sigma_{2N} ^2 \\ &&...&& \\ \sigma_{N1} ^2 & \sigma_{N2}^2 & \dots & \sigma_{NN}^2 \end{bmatrix} \begin{bmatrix} x_1\\\vdots\\x_N \end{bmatrix}=\sum_{i=1}^N\sum_{j=1}^N x_i\sigma_{ij}^2 x_j $$ therefore $$ E[x^T\Sigma x]=\sum_{i=1}^N\sum_{j=1}^N E[x_i\sigma_{ij}^2 x_j] $$ and what should I do in the following?
$$ E[x_i \sigma^2_{ij} x_j] = E[E[x_i \sigma^2_{ij} x_j \mid \sigma^2_{ij}]] = E[(\sigma_{ij}^2)^2] = \text{Var}(\sigma_{ij}^2) + E[\sigma_{ij}^2]^2 $$