# 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}} &= \left[z_{1},z_{2},\ldots,z_{N}\right]^{T} \in {\mathbb{C}}^{N} \\ {\tilde{\bf z}} &= \left[z_{1},z^{*}_{1},\ldots, z_{N},z^{*}_{N}\right]^{T} \in {\mathbb{C}}^{2N} \\ {\hat{\bf z}} &= \left[z^{R}_{1},z^{I}_{1},\ldots, z^{R}_{N},z^{I}_{N}\right]^{T} \in {\mathbb{R}}^{2N}\\ {\bf M} &\in {\mathbb{C}}^{N \times N}\cr \tilde{{\bf M}} & \in {\mathbb{C}}^{2N \times 2N} \cr {\hat{\bf M}} & \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$$\begin{equation} \mathbf{M} = \begin{bmatrix} z_{11} & z_{12} & \dots & z_{1N}\\ z_{21} & z_{22} & \dots & z_{2N}\\ \vdots & \vdots & \ddots & \vdots\\ z_{N1} & z_{N2} & \dots & z_{NN} \end{bmatrix} \end{equation}$$

and the paper later clarifies

$$\begin{equation} \hat{\mathbf{M}} = \begin{bmatrix} z_{11}^R & -z_{11}^I & \dots & z_{1N}^R & - z_{1N}^I\\ z_{11}^I & z_{11}^R & \dots & z_{1N}^I & z_{1N}^R\\ \vdots & \vdots & \ddots & \vdots & \vdots\\ z_{N1}^R & -z_{N1}^I & \dots & z_{NN}^R & - z_{NN}^I\\ z_{N1}^I & z_{N1}^R & \dots & z_{NN}^I & z_{NN}^R\\ \end{bmatrix} \end{equation}$$

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
$$\begin{bmatrix}m_{kl}&0\\0&m_{kl}^*\end{bmatrix}$$
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
$$\begin{bmatrix}m_{kl}^R&-m_{kl}^I\\m_{kl}^I&m_{kl}^R\end{bmatrix}$$
where the superscripts $$^R$$ and $$^I$$ denote real and imaginary parts, respectively.