Say I have the following $$\min_{\bf x_1,x_2}\left\{\|{\bf Ax_1 - Bx_2}\|_2^2 + \text{other terms linear in } \bf x_1,x_2\right\}$$ for matrices $\bf A,B$ and vectors $\bf x_1,x_2$
How to arrange to solve it using linear methods like a linear equation system or linear least squares?
$\| Ax-b\|_2^2 + \| Cx-d\|_2^2 $ is equal to $\| Dx-e\|_2^2$ where $D = \begin{pmatrix}A\\B\end{pmatrix}$ and $e = \begin{pmatrix}b\\d\end{pmatrix}$.
$\| A_1x_1 + A_2 x_2-b\|_2^2$ can be written as $\| Ax-b\|_2^2$ where $A = \begin{pmatrix}A_1 & A_2\end{pmatrix}$ and $x = \begin{pmatrix}x_1\\x_2\end{pmatrix}$