For rank $r$ matrix $A\in\mathbb F^{m\times n}$, a reduced or rank LU factorization uses $A=L_rU_r$ with $L_r$ of shape $m\times r$ and $U_r$ of shape $r\times n$, omitting redundant zero directions. Permutations may be needed: $PA=L_rU_r$. Terminology varies; always state the shapes and pivot conventions.
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