Clarifying matrix notation from an ICA-CMN paper

Question

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!

Answer 1

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.