I have a matrix $X = \begin{bmatrix} 0.5000 + 0.5000i & 0.5000 - 0.5000i\\ 0.5000 - 0.5000i & 0.5000 + 0.5000i \end{bmatrix}$ multiplied with a column containing a complex number and its conjugate, as below:
$y = \begin{bmatrix} y_1\\ y_2 \end{bmatrix} = \begin{bmatrix} 0.5000 + 0.5000i & 0.5000 - 0.5000i\\ 0.5000 - 0.5000i & 0.5000 + 0.5000i \end{bmatrix} \times \begin{bmatrix} s\\ s' \end{bmatrix}$
I am wondering if we can recover $s$ from only $y_1$ or $y_2$. I mean, as long as the vector $s$ is only containing a complex number with its conjugate, so we can estimate $s$ from only $y_1$. I In other words, $s$ is supposed to be included in $y_1$, and also included in $y_2$. Is that right ? But, I don't know how can I estimate $s$ based on only $y_1$.
If $s=s_r+is_i$ and $s'=s_r-is_i$ then (if I did not make any mistake): $$y_1=\frac14(s_r+is_i+is_r-s_i+s_r-is_i-is_r-s_i)=\frac12 (s_r-s_i)\\ y_2=\frac12(s_r+s_i)$$ Notice that neither $y_1$ or $y_2$ alone allows you to find both $s_r$ and $s_i$.