Expectation of vector-matrix-vector product

Question

I have the following term

$$ E(X_t' \Sigma X_t) $$

where $X_t$ is a $n \times 1$ vector of random variables and $\Sigma$ is a $n \times n$ matrix of constants. Is it possible to somehow "factor out" the matrix $\Sigma$ out the expectation?

Answer 1

One way to "factor" out the matrix $\Sigma$ is by using the trace operator, Since $X^T \Sigma X$ can be seen as a $1\times 1$ matrix, you have that $X^T \Sigma X=\mathrm{Tr} (X^T \Sigma X)$ and hence \begin{align*} \mathbb E[X^T \Sigma X] &= \mathbb E[\mathrm{Tr}(X^T \Sigma X)]\\ &= \mathbb E[\mathrm{Tr}(\Sigma XX^T )]\\ &= \mathrm{Tr}(\mathbb E[\Sigma XX^T ])\\ &= \mathrm{Tr}(\Sigma \mathbb E[XX^T ]) \end{align*}