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# Definition of rank factorization of a matrix

Put content here.**Definition:** A **rank factorization** of an $m \times n$ matrix $A$ of rank $r$ is a decomposition:
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$$A = CR$$
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where $C$ is $m \times r$ with full column rank ($r$) and $R$ is $r \times n$ with full row rank ($r$).
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**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 $A$
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**Example:**
$$A = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 1 & 0 & 1 \end{pmatrix} = \begin{pmatrix} 1 & 2 \\ 2 & 4 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \end{pmatrix}$$
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**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: $A^+ = R^+C^+$

# Parents

* Factorization of matrices