# Definition:Triangular Matrix

## Definition

Let $\mathbf T = \begin {bmatrix} a_{1 1} & a_{1 2} & \cdots & a_{1 n} \\ a_{2 1} & a_{2 2} & \cdots & a_{2 n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m 1} & a_{m 2} & \cdots & a_{m n} \\ \end {bmatrix}$ be a matrix of order $m \times n$.

Then $\mathbf T$ is a triangular matrix if and only if all the elements either above or below the diagonal are zero.

### Upper Triangular Matrix

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

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

That is, $\mathbf U$ is upper triangular if and only if:

$\forall a_{ij} \in \mathbf U: i > j \implies a_{ij} = 0$

### Lower Triangular Matrix

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.

That is, $\mathbf L$ is lower triangular if and only if:

$\forall a_{i j} \in \mathbf U: i < j \implies a_{i j} = 0$

## Also defined as

Some sources define a triangular matrix only as a square matrix.