Definition:Triangular Matrix/Lower Triangular Matrix

Definition
A lower triangular matrix is a matrix in which all the upper triangular elements are zero.

That is, all the non-zero elements are on the main diagonal or in the lower triangle:


 * $\mathbf L = \begin{bmatrix}

a_{11} &           0 &            0 & \cdots &                0 & 0 \\ a_{21} &      a_{22} &            0 & \cdots &                0 & 0 \\ a_{31} &      a_{32} &       a_{33} & \cdots &                0 & 0 \\ \vdots &      \vdots &       \vdots & \ddots &           \vdots & \vdots \\ a_{n - 1, 1} & a_{n - 1, 2} & a_{n - 1, 3} & \cdots & a_{n - 1, n - 1} & 0 \\ a_{n1} &      a_{n2} &       a_{n3} & \cdots & a_{n - 1, 1}     & a_{nn} \\ \end{bmatrix}$

That is, $\mathbf L$ is lower triangular :
 * $\forall a_{ij} \in \mathbf U: i < j \implies a_{ij} = 0$

Also defined as
Some sources define a lower triangular matrix only as a square matrix.

Also see

 * Definition:Upper Triangular Matrix