Upper triangular matrix

Definition

Let $R$ be a ring and let $n \ge 0$ be an integer. A square matrix $U = (u_{ij})$ of order $n$ over $R$ is upper triangular when $u_{ij} = 0$ for all $i > j$. It is strictly upper triangular when $u_{ij} = 0$ for all $i \ge j$.

Remarks

Sums and products of upper triangular matrices of the same order are upper triangular. The determinant of a triangular matrix is the product of its diagonal entries. The diagonal entries of an upper triangular matrix over a field are its eigenvalues. A system $Ux = b$ with $U$ upper triangular and invertible is solved by back substitution.

Explanations

L1 human

An upper triangular matrix has zeros everywhere below its diagonal. Its nonzero entries fill a triangle at the top right. For example, $\begin{pmatrix} 1 & 2 \\ 0 & 3 \end{pmatrix}$ is upper triangular.