Matrix decomposition into two arbitrary sized matrices Given a matrix $A$ of dimensions $m\times{}n$, I am interested in decomposing $A$ into the product $BC$ where $B$ is a $m\times{}p$ matrix and $C$ is a $p\times{}n$ matrix.
What are the methods to perform such a decomposition? What are the possible family of solutions? Are these solutions exhaustive?
Background: From an image processing routine, I have an equation $A = BC$ where $A$ is a known $1\times{}12$ vector, $B$ is an unknown $1\times{}6$ vector that contain physical quantities to be recovered, and $C$ is an unknown $6\times{}12$ matrix.
 A: Let us consider the vector case, here are some examples
\begin{align}
  \pmatrix{1 & 0 & 12}\pmatrix{1 \\ 2 \\ 0} &= 1 \\
  \pmatrix{1 & 1 & 2}\pmatrix{0 \\ 1 \\ 0} &= 1 \\
  \pmatrix{0 & 0 & 12}\pmatrix{5 \\ 2 \\ 1/ 12} &= 1 \\
\end{align}
I think you get the idea, generally speaking, taking your $p$ larger and larger can give an infinite set of factorizations.
Arguably the most useful decomposition would be the singular value decomposition. You can write $A = USV^\top$, and take $B=US$ and $C=V^\top$ to have a decomposition into factors that are orthogonal and unitary, respectively. This works for every possible dimension of $m$ and $n$. $p=n$ would be the case in that example.
More generally, to decompose $A= BC$, you can perform row operation on $A$ and obtain any non-singular factor for $B$ (conversely do column operations on $A$ to have any non-singular $C$).
$$A = B\underbrace{B^{-1}A}_{C}\quad\text{for arbitrary non-singular $B$}$$
or
$$A = \underbrace{AC^{-1}}_{B}C\quad\text{for arbitrary non-singular $C$}$$
