I have two matrices ${\bf{X}}_{p\times r}$ and ${\bf{Y}}_{r\times q}$ with $r<\min(p,q)$. Matrix ${\bf Y}$ does not have full row rank (i.e., rank$({\bf Y})<r$). Can I build two new matrices ${\bf{X}}'_{p\times (r-1)}$ and ${\bf{Y'}}_{(r-1)\times q}$ such that the following three conditions hold:
$\bf{X}' \bf{Y}' = \bf{X} \bf{Y}$
$\|\bf{X}'\|_1 \leq \|\bf{X}\|_1$
$\|\bf{Y}'\|_1 \leq \|\bf{Y}\|_1$
EDIT: By $\ell_1$ norm of a matrix, I mean the sum of absolute values of its elements.