Let $\mathbf{w} \sim \mathcal{CN}(\mathbf{0},\mathbf{I})$ be a $N$-dimensional, complex Gaussian random vector with zero mean vector and identity covariance matrix.
I have to evaluate the following matrix: \begin{equation} \mathbf{C} = E\{[\mathrm{vec}(\mathbf{w}\mathbf{w}^H)][\mathrm{vec}(\mathbf{w}\mathbf{w}^H)]^H\} \end{equation} where $E\{\cdot\}$ is the expectation operator, $\mathrm{vec}(\cdot)$ defines the vectorization operator obtained by stacking the columns of a matrix on top of one another, and $H$ indicate the Hermitian operator. Consequently, $[\mathrm{vec}(\mathbf{w}\mathbf{w}^H)]$ is an $N^2$-dimensional complex random vector. Can anyone help me with it?
Thanks!