Definition L = [ l 11 0 0 l 21 l 22 0 l 31 l 32 l 33 ] , L T = [ l 11 l 21 l 31 0 l 22 l 32 0 0 l 33 ]
L =
\begin{bmatrix}
l_{11} & 0 & 0 \\
l_{21} & l_{22} & 0 \\
l_{31} & l_{32} & l_{33}
\end{bmatrix}
,\:
L^T =
\begin{bmatrix}
l_{11} & l_{21} & l_{31} \\
0 & l_{22} & l_{32} \\
0 & 0 & l_{33}
\end{bmatrix}
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L = l 11 l 21 l 31 0 l 22 l 32 0 0 l 33 , L T = l 11 0 0 l 21 l 22 0 l 31 l 32 l 33 Solve L L T X = B for X to solve the system \text{Solve}\: LL^TX = B \: \text{for X to solve the system} Solve L L T X = B for X to solve the system Let L T X = Y \text{Let}\: L^TX = Y Let L T X = Y First solve L Y = B for Y and then solve L T X = Y for X \text{First solve}\: LY = B \: \text{for}\: Y \: \text{and then solve} \: L^TX = Y\: \text{for}\: X First solve L Y = B for Y and then solve L T X = Y for X