Let $x \in \mathbb{C}^{N \times 1}$ be a vector of random variables with $x \sim \mathcal{N}_C (0, \Sigma_{xx})$ and let $A \in \mathbb{C}^{N \times N} $ be a Hermitian matrix (i.e. $A = A^H$ with $(\cdot)^H$ being the conjugate transpose). Through simple element-wise multiplication it can be observed that:
$$ \text{diag}(x)~ A ~\text{diag}(x)^H = A \circ (xx^H)$$ where $\circ$ denotes the Hadamard product. $\text{diag}(x)$ is the diagonal matrix with $x_i$ as its $i$-th diagonal element.
More specifically I am interested to show that
$$ E[ ~\text{diag}(x)~ A ~\text{diag}(x)^H ~] = A \circ \Sigma_{xx} $$ where $E[\cdot]$ is the expectation operator.
Is there any reference I can cite one of the above properties?