# Complex Gaussian random vector, vec operator and expectation

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!

• Your notation E{???} needs clarification. – herb steinberg Feb 9 '18 at 22:40
• 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. – Vuk Feb 10 '18 at 9:08
• You have an expression [vec...][vec...]. What does this mean? – herb steinberg Feb 11 '18 at 17:26
• 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 $[\mathbf{a}\mathbf{a}^H]_{ij}=a_ia_j^*$. – Vuk Feb 11 '18 at 17:51
• 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? – herb steinberg Feb 12 '18 at 19:57