Say $x$ is a random vector in $\mathbb{R}^n$. Then, given a (deterministic) symmetric real positive definite matrix $A$, if we want to calculate the expectation of the quadratic form, we can use the identity $$ E[\langle Ax, x \rangle] = \mathrm{tr} (E A xx^T) = \mathrm{tr} (A \Sigma), $$ where $\Sigma = E[xx^T]$.
Is there an analogue when instead $X$ is random matrix and we look at the matrix quadratic form $L_X \colon A \mapsto X^T A X$? This is evidently a linear map in $A$ but what is the right analogue of $\Sigma$ for $E L_X$? (Of course, one can do this by vectorizing $A$ and using Kronecker products but this loses the matrix structure of the problem. Is there a more natural way to do it?)
