Definition of rank factorization of a matrix

Definition: A rank factorization of an $m \times n$ matrix $A$ of rank $r$ is a decomposition:

$$A = CR$$

where $C$ is $m \times r$ with full column rank ($r$) and $R$ is $r \times n$ with full row rank ($r$).

Construction: If $A$ has rank $r$, then:

• $C$ consists of the pivot columns of $A$ (the linearly independent columns)
• $R$ is obtained from the nonzero rows of the reduced row echelon form of $A$

Example:
$$A = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 1 & 0 & 1 \end{pmatrix} = \begin{pmatrix} 1 & 2 \\ 2 & 4 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \end{pmatrix}$$

Properties:

• The rank factorization reveals that $A$ maps $\mathbb{R}^n$ through an $r$-dimensional intermediate space
• The pseudoinverse can be computed from rank factorization: $A^+ = R^+C^+$