How to convert a $3\times 5$ matrix to a product of a $3\times 2$ and $2\times 5$ matrix?

Question

I have a matrix $$A = \begin{bmatrix} 1 & 1 & 2 & 2 & 3\\ 1 & 0 & 1 & 2 & 2\\ 0 & 1 & 1 & 0 & 1\\ \end{bmatrix},$$ and I would like to write this as a product of a $3\times 2$ and $2\times 5$ matrix.

By reducing the matrix to row echelon form I get the $2\times 5$ matrix $$\begin{bmatrix} 1 & 0 & 1 & 2 & 2\\ 0 & 1 & 1 & 0 & 1 \end{bmatrix},$$ How do I get the $3\times 2$ matrix?

Answer 1

Quoting from Rank factorization:

In practice, we can construct one specific rank factorization $A = CF$ as follows: we can compute $B$, the reduced row echelon form of $A$. Then $C$ is obtained by removing from $A$ all non-pivot columns, and $F$ by eliminating all zero rows of $B$.

In your case, $A$ has rank 2, and you already computed the reduced row echelon form without the zero rows as $$ F = \begin{bmatrix} 1 & 0 & 1 & 2 & 2\\ 0 & 1 & 1 & 0 & 1 \end{bmatrix} $$

$C$ is obtained by picking only the pivot-columns (one and two) from $A$ $$ C = \begin{bmatrix} 1 & 1 \\ 1 & 0\\ 0 & 1\\ \end{bmatrix} $$ and $A=CF$ is the desired decomposition.

Answer 2

Each row of the $3×2$ matrix represents coefficients for a linear combination of the rows of the $2×5$ matrix that produces a row of the original matrix.

Using the found row-echelon form, let's consider the first row of the $3×2$ matrix: $$(1,1,2,2,3)=a(1,0,1,2,2)+b(0,1,1,0,1)$$ We see that $a=b=1$, so the first row is $(1,1)$. Similar computations apply for the other two rows.