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I want to compute the following expectation term:

$E[{\bf{XA}}{{\bf{X}}^T}]$

where ${\bf X} \in R^{M \times M}$ and its elements are normal random variables such that

$vec\left( {\bf{X}} \right)\sim \cal N\left( {\boldsymbol \mu ,\bf \Sigma } \right)$

$\bf A$ is a positive definite matrix with proper dimensions and $vec(.)$ is the vectorization operator. Any hint on how I can derive a nice formula?

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$\newcommand{\X}{\mathbf X} \newcommand{\si}{\sigma}$ Suppose that $A:=(a_{ij})_{i,j=1}^m$ is an $m\times m$ matrix and $\X=(X_{ij})_{i,j=1}^m$ is a random $m\times m$ matrix with $EX_{ij}=\mu_{ij}$ and $Cov(X_{ij},X_{kl})=\si_{ij,kl}$. Then the $il$-entry of the matrix $E\X A\X^T$ is
$$(E\X A\X^T)_{il}=\sum_{j,k}EX_{ij}a_{jk}X_{lk} =\sum_{j,k}a_{jk}(\mu_{ij}\mu_{kl}+\si_{ij,lk}) =(MAM^T)_{il}+\sum_{j,k}a_{jk}\si_{ij,lk}, $$ where $M:=(\mu_{ij})_{i,j=1}^m$. So, $$E\X A\X^T=MAM^T+R, $$ where $R:=(r_{il})_{i,l=1}^m$ with $r_{il}:=\sum_{j,k}a_{jk}\si_{ij,lk}$.

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