Definition L = [ l 11 0 0 l 21 l 22 0 l 31 l 32 l 33 ] , U = [ u 11 u 12 u 13 0 u 22 u 23 0 0 u 33 ]
L =
\begin{bmatrix}
l_{11} & 0 & 0 \\
l_{21} & l_{22} & 0 \\
l_{31} & l_{32} & l_{33}
\end{bmatrix}
,\:
U =
\begin{bmatrix}
u_{11} & u_{12} & u_{13} \\
0 & u_{22} & u_{23} \\
0 & 0 & u_{33}
\end{bmatrix}
\\[5px]
L = l 11 l 21 l 31 0 l 22 l 32 0 0 l 33 , U = u 11 0 0 u 12 u 22 0 u 13 u 23 u 33 Solve L U X = B for X to solve the system \text{Solve}\: LUX = B \: \text{for X to solve the system} Solve LU X = B for X to solve the system Let U X = Y \text{Let}\: UX = Y Let U X = Y First solve L Y = B for Y and then solve U X = Y for X \text{First solve}\: LY = B \: \text{for}\: Y \: \text{and then solve} \: UX = Y\: \text{for}\: X First solve L Y = B for Y and then solve U X = Y for X