I am reading this communication article:
I am confused as to how to write these notation to prove the properties, for example if I take A to be a $2\times2$ matrix like: $$ \mathbf C=\begin{bmatrix}(x_1+jy_1) \ (x_2+jy_2) \\ (x_3+jy_3) \ (x_4+jy_4) \end{bmatrix}$$ now how to write this in the real and imaginary matrix form. I tried to do by adding all real and imaginary parts at the given locations respectively.But that doesn't prove the properties.
At first glance, if $z \in \mathbb{C}^n$ then $\hat{z} \in \mathbb{R}^{2n}$. Similarly, $A \in \mathbb{C}^{n \times m}$ then $\hat{A} \in \mathbb{R}^{2n \times 2m}$ since $\Re(A), \Im(A)$ will be a $n \times m$ matrix.
So for your $\mathbf{C}$:
$$\hat{\mathbf{C}} = \left [ \begin{array}{rr} \left ( \begin{array}{rr} x_1 & x_2 \\ x_3 & x_4 \end{array} \right ) & \left ( \begin{array}{rr} -y_1 & -y_2 \\ -y_3 & -y_4 \end{array} \right ) \\ \left ( \begin{array}{rr} y_1 & y_2 \\ y_3 & y_4 \end{array} \right ) & \left ( \begin{array}{rr} x_1 & x_2 \\ x_3 & x_4 \end{array} \right ) \end{array} \right ]$$