Definition:Triangular Matrix

Definition

Let $\mathbf T = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \\ \end{bmatrix}$ be a square matrix of order $n$.

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

Upper Triangular Matrix

An upper triangular matrix is one in which all the elements below the diagonal are zero. That is, all the non-zero elements are in the upper triangle:

$\mathbf U = \begin{bmatrix} a_{11} & a_{12} & a_{13} & \cdots & a_{1,n-1} & a_{1n} \\ 0 & a_{22} & a_{23} & \cdots & a_{2,n-1} & a_{2n} \\ 0 & 0 & a_{33} & \cdots & a_{3,n-1} & a_{3n} \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & a_{n-1,n-1} & a_{n-1,n} \\ 0 & 0 & 0 & \cdots & 0 & a_{nn} \\ \end{bmatrix}$

That is, $\mathbf U$ is upper triangular iff $\forall a_{ij} \in \mathbf U: i > j \implies a_{ij} = 0$.

Lower Triangular Matrix

A lower triangular matrix is one in which all the elements above the diagonal are zero. That is, all the non-zero elements are 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 iff $\forall a_{ij} \in \mathbf U: i < j \implies a_{ij} = 0$.

Transpose

The transpose of an upper triangular matrix is clearly a lower triangular matrix, and vice versa.