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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 $y_1$.

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  • $\begingroup$ Hint : invert the matrix X. $\endgroup$ Commented Feb 21, 2022 at 14:35
  • $\begingroup$ @KurtG. I mean recovering $s$ based only on $y_1$ OR $y_2$ $\endgroup$ Commented Feb 21, 2022 at 14:54

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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 $s_i$.

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  • $\begingroup$ I think $y_1 = s_r - s_i$ and $y_2 = s_r+s_i$. thank you $\endgroup$ Commented Feb 22, 2022 at 1:37

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