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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!

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  • $\begingroup$ Your notation E{???} needs clarification. $\endgroup$ – herb steinberg Feb 9 '18 at 22:40
  • $\begingroup$ Here some additional clarification. $E\{\cdot\}$ 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. $\endgroup$ – Vuk Feb 10 '18 at 9:08
  • $\begingroup$ You have an expression [vec...][vec...]. What does this mean? $\endgroup$ – herb steinberg Feb 11 '18 at 17:26
  • $\begingroup$ The expression $[\mathrm{vec}(\mathbf{w}\mathbf{w}^H)][\mathrm{vec}(\mathbf{w}\mathbf{w}^H)]^H$ represents the $N^2 \times N^2$ matrix obtained by the standard (row-column) matrix product of the vectors $\mathrm{vec}(\mathbf{w}\mathbf{w}^H)$ and $[\mathrm{vec}(\mathbf{w}\mathbf{w}^H)]^H$. In particular, let $\mathbf{a}$ be a complex column vector, $\mathbf{a}\mathbf{a}^H$ represents the matrix whose entries are defined as $[\mathbf{a}\mathbf{a}^H]_{ij}=a_ia_j^*$. $\endgroup$ – Vuk Feb 11 '18 at 17:51
  • $\begingroup$ I don't think I can help you further. I have never encountered an expectation for a matrix. Could it mean expectation for each element? $\endgroup$ – herb steinberg Feb 12 '18 at 19:57

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