I have a real matrix $X$ whose size is $N \times N$, this matrix is multiplied with a real vector $s$ with size $N \times 1$, assuming the form of the resulted vector $y$ is known $y = X \times s = \begin{bmatrix}0 \\ y_1 \\ 0 \\ y_2 \end{bmatrix}$ , which is means it's sparse with known location of sparsity, we assumed here that $N = 4$.
the same matrix $X$ is multiplied with another vector $S$ and the resulted matrix $z$ has the same form of $y$ which is $z = X \times S = \begin{bmatrix}0 \\ z_1 \\ 0 \\ z_2 \end{bmatrix}$ . If we built the $q$ based on both vectors $y$ and $z$ such that $q = \begin{bmatrix}y_1 \\ z_1 \\ y_2 \\ z_2 \end{bmatrix}$, or for example $q = \begin{bmatrix}y_1 \\ y_2 \\ z_1 \\ z_2 \end{bmatrix}$ which means we eliminated the zeros.
My questions, how can I also build the matrix $\mathbb{X}$ equivalent to the vector $q$? I mean mix the two multiplication in one operation such that the result should give the vector $q$. It can be for example: $\mathbb{X} \times \begin{bmatrix} s \\ S \end{bmatrix} = q $
Let $P = \begin{bmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \end{bmatrix}$. This is a permutation matrix. Thus, we have that
$$ Pz = \begin{bmatrix} z_1 \\ 0 \\ z_2 \\ 0 \end{bmatrix}. $$
Now $$q = Xs + PXS = \underbrace{\begin{bmatrix} X & PX\end{bmatrix}}_{\mathbb{X}} \begin{bmatrix} s \\ S\end{bmatrix}.$$