Definition of a lower triangular matrix

Definition. An $n \times n$ matrix $A = (a_{ij})$ is called lower triangular if all entries above the main diagonal are zero, i.e., $a_{ij} = 0$ for all $i < j$.

A lower triangular matrix has the form:
$$A = \begin{pmatrix} a_{11} & 0 & 0 & \cdots & 0 \\ a_{21} & a_{22} & 0 & \cdots & 0 \\ a_{31} & a_{32} & a_{33} & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & a_{n3} & \cdots & a_{nn} \end{pmatrix}$$

Example. $\begin{pmatrix} 2 & 0 & 0 \\ -1 & 3 & 0 \\ 4 & 1 & 5 \end{pmatrix}$ is lower triangular.

Note: $A$ is lower triangular if and only if $A^T$ is upper triangular.