How to find the set of solutions of $Ax=By$ for $A,B$ matrices?

Question

Let $A,B$ be arbitrary matrices with the same number of rows.

How can we find the set of solutions $x,y$ to the matrix equation $Ax=By$?

I understand that this problem is probably related to that of finding a basis for the intersection of two vector spaces, which can be solved as shown in the answers to this question. However, the methods outlined there work when $A,B$ have as columns orthonormal sets, and therefore $\operatorname{Ker}(A)=\operatorname{Ker}(B)=\{0\}$, which needs not be the case here.

How is this kind of equation solved in the general case?

Answer 1

construct the matrix (using blockmatrices): $$ M = \begin{pmatrix} A& -B\\ \end{pmatrix} $$ and solve the system $Mz = 0$ where $z = \begin{pmatrix} x_1\\ \vdots\\ x_n\\ y_1\\ \vdots\\ y_m\end{pmatrix}$.

Answer 2

The problem makes sense only if $A$ is $n\times m$ and $B$ is $n\times k$, i.e., $A$ and $B$ must have the same number of rows. Let $C$ be the $n\times(m+k)$ matrix obtained by juxtaposing $A$ and $-B$. Then The solutions $z$ of $Cz=0$ correspond to solutions of the original claim, namely the top $m$ components are your $x$ and the bottom $k$ components are $y$.

Answer 3

You can solve for the column vector x if the column vector $y$ is given.

Once you have $y$ you have the vector $b=By$ and solve the system $Ax=b$ For $x$