Clarifying matrix notation from an ICA-CMN paper
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.