Clarifying matrix notation from an ICA-CMN paper In the paper I am referring (and here from citeseer), complex vectors $\mathbf{z}$ and matrix $\mathbf{M}$ were defined as follows

\begin{align}
{{\bf z}} &amp;= \left[z_{1},z_{2},\ldots,z_{N}\right]^{T} \in {\mathbb{C}}^{N} \\
{\tilde{\bf z}} &amp;= \left[z_{1},z^{*}_{1},\ldots, z_{N},z^{*}_{N}\right]^{T} \in {\mathbb{C}}^{2N} \\
{\hat{\bf z}} &amp;= \left[z^{R}_{1},z^{I}_{1},\ldots, z^{R}_{N},z^{I}_{N}\right]^{T} \in {\mathbb{R}}^{2N}\\
{\bf M} &amp;\in {\mathbb{C}}^{N \times N}\cr \tilde{{\bf M}} &amp; \in {\mathbb{C}}^{2N \times 2N} \cr {\hat{\bf M}} &amp; \in {\mathbb{R}}^{2N \times 2N} 
\end{align}

However, I can't seem to figure out how $\tilde{\mathbf{M}}$ would look like.
I have\begin{equation}
\mathbf{M} =
    \begin{bmatrix}
    z_{11} &amp; z_{12}   &amp; \dots     &amp; z_{1N}\\
    z_{21} &amp; z_{22}   &amp; \dots     &amp; z_{2N}\\
    \vdots &amp; \vdots     &amp; \ddots    &amp; \vdots\\
    z_{N1} &amp; z_{N2} &amp; \dots         &amp; z_{NN}
    \end{bmatrix}
\end{equation}
and the paper later clarifies
\begin{equation}
\hat{\mathbf{M}} =
    \begin{bmatrix}
    z_{11}^R &amp; -z_{11}^I &amp; \dots    &amp; z_{1N}^R &amp; - z_{1N}^I\\
    z_{11}^I &amp; z_{11}^R &amp; \dots    &amp; z_{1N}^I &amp;  z_{1N}^R\\
    \vdots   &amp; \vdots    &amp; \ddots   &amp; \vdots &amp; \vdots\\
    z_{N1}^R &amp; -z_{N1}^I &amp; \dots    &amp; z_{NN}^R &amp; - z_{NN}^I\\
    z_{N1}^I &amp; z_{N1}^R &amp; \dots    &amp; z_{NN}^I &amp;  z_{NN}^R\\
    \end{bmatrix}
\end{equation}
but not for  $\tilde{\mathbf{M}}$.
Thank you very much!
The matrices $\hat{\mathbf{M}}$ and $\tilde{\mathbf{M}}$ are constructed in such a way that the relation $\mathbf{M}\mathbf{x}=\mathbf{y}$ implies $\hat{\mathbf{M}}\hat{\mathbf{x}}=\hat{\mathbf{y}}$ and $\tilde{\mathbf{M}}\tilde{\mathbf{x}}=\tilde{\mathbf{y}}$. 

Consequently, for constructing the matrix $\tilde{\mathbf{M}}$,  each element $m_{kl}$ of $\mathbf{M}$ must be replaced by a $2\times 2$ sub-matrix

$$\begin{bmatrix}m_{kl}&amp;0\\0&amp;m_{kl}^*\end{bmatrix}$$

In a similar way, the matrix $\hat{\mathbf{M}}$ is formed by replacing each element $m_{kl}$ of $\mathbf{M}$ by the $2\times 2$ sub-matrix

$$\begin{bmatrix}m_{kl}^R&amp;-m_{kl}^I\\m_{kl}^I&amp;m_{kl}^R\end{bmatrix}$$

where the superscripts $^R$ and $^I$ denote real and imaginary parts, respectively.