Es una factorización LU de matrices simétricas $$ \begin{equation*} \mathbf{A} = \mathbf{L} \ \mathbf{L}^{T} \end{equation*} $$
Factorizar la matriz simétrica \( \mathbf{A}_{m \times m} \) $$ \begin{equation*} \begin{bmatrix} a_{11} & & & \\ a_{21} & a_{22} & & \\ a_{31} & a_{32} & a_{33} & \\ a_{41} & a_{42} & a_{43} & a_{44} \end{bmatrix} = \begin{bmatrix} l_{11} & 0 & 0 & 0 \\ l_{21} & l_{22} & 0 & 0 \\ l_{31} & l_{32} & l_{33} & 0 \\ l_{41} & l_{42} & l_{43} & l_{44} \end{bmatrix} \begin{bmatrix} l_{11} & l_{21} & l_{31} & l_{41} \\ 0 & l_{22} & l_{32} & l_{42} \\ 0 & 0 & l_{33} & l_{43} \\ 0 & 0 & 0 & l_{44} \end{bmatrix} \end{equation*} $$
Multiplicando el lado derecho $$ \begin{equation*} \begin{bmatrix} a_{11} & & & \\ a_{21} & a_{22} & & \\ a_{31} & a_{32} & a_{33} & \\ a_{41} & a_{42} & a_{43} & a_{44} \end{bmatrix} = \begin{bmatrix} l_{11}^{2} & & & \\ l_{11} l_{21} & l_{21}^{2} + l_{22}^{2} & & \\ l_{11} l_{31} & l_{21} l_{31} + l_{22} l_{32} & l_{31}^{2} + l_{32}^{2} + l_{33}^{2} & \\ l_{11} l_{41} & l_{21} l_{41} + l_{22} l_{42} & l_{31} l_{41} + l_{32} l_{42} + l_{33} l_{43} & l_{41}^{2} + l_{42}^{2} + l_{43}^{2} + l_{44}^{2} \end{bmatrix} \end{equation*} $$
Despejando \( \mathbf{L} \) $$ \begin{equation} \begin{bmatrix} l_{11} & & & \\ l_{21} & l_{22} & & \\ l_{31} & l_{32} & l_{33} & \\ l_{41} & l_{42} & l_{43} & l_{44} \end{bmatrix} = \begin{bmatrix} \sqrt{a_{11}} & & & \\ \frac{a_{21}}{l_{11}} & \sqrt{a_{22} - l_{21}^{2}} & & \\ \frac{a_{31}}{l_{11}} & \frac{a_{32} - l_{31} l_{21}}{l_{22}} & \sqrt{a_{33} - l_{31}^{2} - l_{32}^{2}} & \\ \frac{a_{41}}{l_{11}} & \frac{a_{42} - l_{41} l_{21}}{l_{22}} & \frac{a_{43} - l_{41} l_{31} - l_{42} l_{32}}{l_{33}} & \sqrt{a_{44} - l_{41}^{2} - l_{42}^{2} - l_{43}^{2}} \end{bmatrix} \tag{11.1} \end{equation} $$