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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.

Ang Zhi Ping
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  • Is $C$ known also then? This would be a least squares problem, and solvable using the SVD or functionality from say Matlab/Octave. – adam W Mar 28 '13 at 03:47
  • $C$ is also unknown. Updated the problem background. – Ang Zhi Ping Mar 28 '13 at 04:00
  • Way too many possibilities, you need some more constraints on the problem. For instance $A = \pmatrix{1 & 0 & 0 & \cdots} \pmatrix{A \ 0 \ 0 \ \vdots}$ – adam W Mar 28 '13 at 04:11

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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 $$A = \underbrace{AC^{-1}}_{B}C\quad\text{for arbitrary non-singular $C$}$$

adam W
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