Suppose that $A_{2n\times 2k},B_{2k\times 2k},C_{2n\times 2n},D_{2n\times 2k}$ are matrices with $k\geq n$. Both $A,C$ are known and $C$ is upper triangular.
I'm trying to find $B,D$ such that $AB=CD$, $B$ is upper triangular and $D$ is a block matrix which looks like $$ \begin{bmatrix} D'_{n\times k} & 0\\ 0 & D''_{n\times k} \end{bmatrix}. $$
My feeling is that $B,C$ are transformations that convert $A$ to $D$, with $C$ known. I checked some particular cases but I don't know whether there exists a solution (or solutions) in general.