Definition:Triangular Matrix/Upper Triangular Matrix

Definition
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:


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

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

Also see

 * Definition:Lower Triangular Matrix