# LU Factorization Any matrix $A \in \mathbb{C}^{n \times n}$ has an LU factorization $P A=L U$, where $P$ is a permutation matrix, $L$ is unit lower triangular (lower triangular with 1 s on the diagonal), and $U$ is upper triangular. For example: $ \left[\begin{array}{rrrr} 3 & -1 & 1 & 1 \\ -1 & 3 & 1 & -1 \\ -1 & -1 & 3 & 1 \\ 1 & 1 & 1 & 3 \end{array}\right]=\left[\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ -\frac{1}{3} & 1 & 0 & 0 \\ -\frac{1}{3} & -\frac{1}{2} & 1 & 0 \\ \frac{1}{3} & \frac{1}{2} & 0 & 1 \end{array}\right]\left[\begin{array}{rrrr} 3 & -1 & 1 & 1 \\ 0 & \frac{8}{3} & \frac{4}{3} & -\frac{2}{3} \\ 0 & 0 & 4 & 1 \\ 0 & 0 & 0 & 3 \end{array}\right] $ --- ## References