La factorización o descomposición LU, factoriza la matriz $\mathbf{A}$ como un producto de la matriz triangular inferior $\mathbf{L}$ y la matriz triangular superior $\mathbf{U}$
\begin{equation*} \mathbf{A} = \mathbf{L} \ \mathbf{U} \end{equation*}Multiplicando
\begin{equation*} \begin{bmatrix} l_{11} u_{11} & l_{11} u_{12} & l_{11} u_{13} & l_{11} u_{14} \\ l_{21} u_{11} & l_{21} u_{12} + l_{22} u_{22} & l_{21} u_{13} + l_{22} u_{23} & l_{21} u_{14} + l_{22} u_{24} \\ l_{31} u_{11} & l_{31} u_{12} + l_{32} u_{22} & l_{31} u_{13} + l_{32} u_{23} + l_{33} u_{33} & l_{31} u_{14} + l_{32} u_{24} + l_{33} u_{34} \\ l_{41} u_{11} & l_{41} u_{12} + l_{42} u_{22} & l_{41} u_{13} + l_{42} u_{23} + l_{43} u_{33} & l_{41} u_{14} + l_{42} u_{24} + l_{43} u_{34} + l_{44} u_{44} \end{bmatrix} = \begin{bmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{21} & a_{22} & a_{23} & a_{24} \\ a_{31} & a_{32} & a_{33} & a_{34} \\ a_{41} & a_{42} & a_{43} & a_{44} \end{bmatrix} \end{equation*}Usando un cambio de variable
\begin{equation*} \begin{bmatrix} l_{11} u_{11} & l_{11} u_{12} & l_{11} u_{13} & l_{11} u_{14} \\ l_{21} u_{11} & a_{11}' & a_{12}' & a_{13}' \\ l_{31} u_{11} & a_{21}' & a_{22} & a_{23}' \\ l_{41} u_{11} & a_{31}' & a_{32}' & a_{33}' \end{bmatrix} = \begin{bmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{21} & a_{22} & a_{23} & a_{24} \\ a_{31} & a_{32} & a_{33} & a_{34} \\ a_{41} & a_{42} & a_{43} & a_{44} \end{bmatrix} \end{equation*}Factorizando la submatriz $\mathbf{A}'$
\begin{equation*} \begin{bmatrix} l_{11}' & 0 & 0 \\ l_{21}' & l_{22}' & 0 \\ l_{31}' & l_{32}' & l_{33}' \end{bmatrix} \begin{bmatrix} u_{11}' & u_{12}' & u_{13}' \\ 0 & u_{22}' & u_{23}' \\ 0 & 0 & u_{33}' \end{bmatrix} = \begin{bmatrix} a_{11}' & a_{12}' & a_{13}' \\ a_{21}' & a_{22}' & a_{23}' \\ a_{31}' & a_{32}' & a_{33}' \end{bmatrix} \end{equation*}Multiplicando
\begin{equation*} \begin{bmatrix} l_{11}' u_{11}' & l_{11}' u_{12}' & l_{11}' u_{13}' \\ l_{21}' u_{11}' & l_{21}' u_{12}' + l_{22}' u_{22}' & l_{21}' u_{13}' + l_{22}' u_{23}' \\ l_{31}' u_{11}' & l_{31}' u_{12}' + l_{32}' u_{22}' & l_{31}' u_{13}' + l_{32}' u_{23}' + l_{33}' u_{33}' \end{bmatrix} = \begin{bmatrix} a_{11}' & a_{12}' & a_{13}' \\ a_{21}' & a_{22}' & a_{23}' \\ a_{31}' & a_{32}' & a_{33}' \end{bmatrix} \end{equation*}Usando un cambio de variable
\begin{equation*} \begin{bmatrix} l_{11} u_{11} & l_{11} u_{12} & l_{11} u_{13} \\ l_{21} u_{11} & a_{11}'' & a_{12}'' \\ l_{31} u_{11} & a_{21}'' & a_{22}'' \end{bmatrix} = \begin{bmatrix} a_{11}' & a_{12}' & a_{13}' \\ a_{21}' & a_{22}' & a_{23}' \\ a_{31}' & a_{32}' & a_{33}' \end{bmatrix} \end{equation*}Factorizando la submatriz $\mathbf{A}''$
\begin{equation*} \begin{bmatrix} l_{11}'' & 0 \\ l_{21}'' & l_{22}'' \end{bmatrix} \begin{bmatrix} u_{11}'' & u_{12}'' \\ 0 & u_{22}'' \end{bmatrix} = \begin{bmatrix} a_{11}'' & a_{12}'' \\ a_{21}'' & a_{22}'' \end{bmatrix} \end{equation*}Multiplicando
\begin{equation*} \begin{bmatrix} l_{11}'' u_{11}'' & l_{11}'' u_{12}'' \\ l_{21}'' u_{11}'' & l_{21}'' u_{12}'' + l_{22}'' u_{22}'' \end{bmatrix} = \begin{bmatrix} a_{11}'' & a_{12}'' \\ a_{21}'' & a_{22}'' \end{bmatrix} \end{equation*}Si $l_{11}'' = 1$ y $l_{22}'' = 1$
\begin{equation*} \begin{bmatrix} u_{11}'' & u_{12}'' \\ l_{21}'' u_{11}'' & l_{21}'' u_{12}'' + u_{22}'' \end{bmatrix} = \begin{bmatrix} a_{11}'' & a_{12}'' \\ a_{21}'' & a_{22}'' \end{bmatrix} \end{equation*}Despejando
\begin{align*} u_{11}'' &= a_{11}'' \\ u_{12}'' &= a_{12}'' \\ l_{21}'' &= \frac{a_{21}''}{u_{11}''} = \frac{a_{21}''}{a_{11}''} \\ u_{22}'' &= a_{22}'' - l_{21}'' u_{12}'' = a_{22}'' - \frac{a_{21}''}{a_{11}''} a_{12}'' \end{align*}Reemplazando
\begin{equation*} \begin{bmatrix} 1 & 0 \\ \frac{a_{21}''}{a_{11}''} & 1 \end{bmatrix} \begin{bmatrix} a_{11}'' & a_{12}'' \\ 0 & a_{22}'' - \frac{a_{21}''}{a_{11}''} a_{12}'' \end{bmatrix} = \begin{bmatrix} a_{11}'' & a_{12}'' \\ a_{21}'' & a_{22}'' \end{bmatrix} \end{equation*}La matriz triangular superior es la misma que se obtiene mediante una eliminación hacia adelante
\begin{equation} \begin{bmatrix} 1 & 0 & 0 & 0 \\ \frac{a_{21}}{a_{11}} & 1 & 0 & 0 \\ \frac{a_{31}}{a_{11}} & \frac{a_{32}'}{a_{22}'} & 1 & 0 \\ \frac{a_{41}}{a_{11}} & \frac{a_{42}'}{a_{22}'} & \frac{a_{43}''}{a_{33}''} & 1 \end{bmatrix} \begin{bmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\ 0 & a_{22}' & a_{23}' & a_{24}' \\ 0 & 0 & a_{33}'' & a_{34}'' \\ 0 & 0 & 0 & a_{44}''' \end{bmatrix} = \begin{bmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{21} & a_{22} & a_{23} & a_{24} \\ a_{31} & a_{32} & a_{33} & a_{34} \\ a_{41} & a_{42} & a_{43} & a_{44} \end{bmatrix} \end{equation}Si $u_{11}'' = 1$ y $u_{22}'' = 1$
\begin{equation*} \begin{bmatrix} l_{11}'' & l_{11}'' u_{12}'' \\ l_{21}'' & l_{21}'' u_{12}'' + l_{22}'' \end{bmatrix} = \begin{bmatrix} a_{11}'' & a_{12}'' \\ a_{21}'' & a_{22}'' \end{bmatrix} \end{equation*}Despejando
\begin{align*} l_{11}'' &= a_{11}'' \\ u_{12}'' &= \frac{a_{12}''}{l_{11}''} = \frac{a_{12}''}{a_{11}''} \\ l_{21}'' &= a_{21}'' \\ l_{22}'' &= a_{22}'' - l_{21}'' u_{12}'' = a_{22}'' - a_{21}'' \frac{a_{12}''}{a_{11}''} \end{align*}Reemplazando
\begin{equation*} \begin{bmatrix} a_{11}'' & 0 \\ a_{12}'' & a_{22}'' - \frac{a_{21}''}{a_{11}''} a_{12}'' \end{bmatrix} \begin{bmatrix} 1 & \frac{a_{21}''}{a_{11}''} \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} a_{11}'' & a_{12}'' \\ a_{21}'' & a_{22}'' \end{bmatrix} \end{equation*}La matriz triangular inferior transpuesta es la misma que se obtiene mediante una eliminación hacia adelante
\begin{equation} \begin{bmatrix} a_{11} & 0 & 0 & 0 \\ a_{12} & a_{22}' & 0 & 0 \\ a_{13} & a_{23}' & a_{33}'' & 0 \\ a_{14} & a_{24}' & a_{34}'' & a_{44}''' \end{bmatrix} \begin{bmatrix} 1 & \frac{a_{21}}{a_{11}} & \frac{a_{31}}{a_{11}} & \frac{a_{41}}{a_{11}} \\ 0 & 1 & \frac{a_{32}'}{a_{22}'} & \frac{a_{42}'}{a_{22}'} \\ 0 & 0 & 1 & \frac{a_{43}''}{a_{33}''} \\ 0 & 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{21} & a_{22} & a_{23} & a_{24} \\ a_{31} & a_{32} & a_{33} & a_{34} \\ a_{41} & a_{42} & a_{43} & a_{44} \end{bmatrix} \end{equation}