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.

$$\left[\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ l_{21}& 1& 0\\ l_{31}& l_{32}& 1\end{array}\right]\times [\begin{array}{ccc}{u}_{11}& u_{12}& u_{13}\\ 0& u_{22}& u_{23}\\ 0& 0& u_{33}\end{array}$$ ...