I want to compute the following expectation term:


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?


$\newcommand{\X}{\mathbf X} \newcommand{\si}{\sigma}$ 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 $r_{il}:=\sum_{j,k}a_{jk}\si_{ij,lk}$.

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.