Lower Upper Triangular Factorizations

Why

If a system $(A, b)$ is ordinarily reducible, then there exists $L$ unit lower triangular and $U$ upper triangular so that $A = LU$. When does such a factorization exist?

Definition

Let $A \in \R ^{m \times m}$. A lower upper triangular factorization of $A$ is a pair of matrices $(L \in \R ^{m \times m}, U \in \R ^{m \times m})$ where $L$ is unit lower triangular, $U$ is upper triangular and $A = LU$. Other terminology includes lower upper triangular decomposition, LU factorization, and LU decomposition.

What about an $LU$-factorization when $(A, b)$ is not ordinarily reducible? The main issue is that we may encounter a diagonal entry of some reduction of $A$ which is zero.