# Hadamard product decomposition with lower rank matrices

Given integers $$k$$ and $$l$$ and a matrix $$A$$ of rank $$kl$$, can we always find a matrix $$B$$ of rank $$k$$ and a matrix $$C$$ of rank $$l$$, such that $$A$$ is the Hadamard product of $$B$$ and $$C$$, namely $$A=B \odot C$$?

For example, when $$k=2$$ and $$l=2$$ and $$A$$ is the 4 by 4 identity matrix,

$$\begin{equation*} A = \begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\\ \end{bmatrix} \end{equation*}$$

we can find $$B$$ of rank 2 and $$C$$ of rank 2:

$$\begin{equation*} B = \begin{bmatrix} 1 & 1 & 0 & 0\\ 1 & 1 & 0 & 0\\ 0 & 0 & 1 & 1\\ 0 & 0 & 1 & 1\\ \end{bmatrix}, C = \begin{bmatrix} 1 & 0 & 1 & 0\\ 0 & 1 & 0 & 1\\ 1 & 0 & 1 & 0\\ 0 & 1 & 0 & 1\\ \end{bmatrix} \end{equation*}$$

satisfying $$A=B \odot C$$.

Can we find $$B$$ and $$C$$ in the general case for any given $$A$$ and $$k$$, $$l$$?

• I'm confused by the close vote -- is there an obvious solution to this problem that shows it's not research-level? I might be missing something, but I don't see an obvious solution, and some recent research has considered problems like this (e.g., arxiv.org/abs/1812.01449). Apr 23 at 13:41