Lower triangular matrix

Definition

Let $R$ be a ring and let $n \ge 0$ be an integer. A square matrix $L = (l_{ij})$ of order $n$ over $R$ is lower triangular when $l_{ij} = 0$ for all $i < j$. It is strictly lower triangular when $l_{ij} = 0$ for all $i \le j$.

Remarks

A matrix is lower triangular if and only if its transpose is upper triangular. Sums and products of lower triangular matrices of the same order are lower triangular. A system $Lx = b$ with $L$ lower triangular and invertible is solved by forward substitution. In an LU decomposition, the factor $L$ is lower triangular.

Explanations

L1 human

A lower triangular matrix has zeros everywhere above its diagonal. Its nonzero entries fill a triangle at the bottom left. For example, $\begin{pmatrix} 1 & 0 \\ 2 & 3 \end{pmatrix}$ is lower triangular.