Challenge: Lower-upper Decomposition Algorithm

LU Decomposition

Lower-upper (LU) decomposition is the process of factoring an invertible matrix, $A$, to a product of a lower triangular matrix, $L$, and an upper triangular matrix $U$.

$$A = LU$$

An input matrix that’s square and invertible will always have a unique LU decomposition where the $L$ matrix has $1s$ at its diagonal.

$$\begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ l_{21} & 1 & 0 \\ l_{31} & l_{32} & 1 \end{bmatrix} \times \begin{bmatrix} u_{11} & u_{12} & u_{13} \\ 0 & u_{22} & u_{23} \\ 0 & 0 & u_{33} \end{bmatrix}$$ ...