General matrix decomposition

Question

Let $A \in {\mathbb C}^{m \times n}$. Does there exist a decomposition $A=BC$ for every $k \in \mathbb N$ such that matrix $B$ is $m \times k$ and matrix $C$ is $k \times n$?

Answer 1

We can write $A = BC$ with $B,C$ of sizes $m \times k$ and $k \times n$ if and only if $\operatorname{rank}(A) \leq k$.

If $\operatorname{rank}(A) = r \leq k$, then $A$ has a rank factorization $FG$ where $F$ is $m \times r$ and $G$ is $r \times k$. It follows that we can take $$ B = \pmatrix{F & 0_{m \times (k-r)}}, \quad C = \pmatrix{G\\ 0_{(k-r) \times n}}. $$ On the other hand, if $A = BC$ and $B$ has size $m \times k$, then it follows that $$ \operatorname{rank}(A) = \operatorname{rank}(BC) \leq \min\{\operatorname{rank}(B),\operatorname{rank}(C)\} \leq k. $$