Superior Triangular Matrix

• A matrix is triangular superior if all elements below the main diagonal are zero.
• When solving a system of equations using Gauss elimination with a superior triangular matrix, no calculations are needed below the diagonal.

Description

• A matrix in the form: $$ \begin{pmatrix} a_{11} & a_{12} & a_{13} & \cdots & a_{1n} \ 0 & a_{22} & a_{23} & \cdots & a_{2n} \ 0 & 0 & a_{33} & \cdots & a_{3n} \ \vdots & \vdots & \vdots & \ddots & \vdots \ 0 & 0 & 0 & \cdots & a_{nn} \end{pmatrix} $$ is considered a superior triangular matrix.
• Formally, matrix $A \in M_{n \times n} (\mathbb{F})$ is superior triangular if: $a_{ij} = 0, \forall i > j$
• A matrix that is both triangular and symmetric is diagonal.

Example

• This matrix is superior triangular: $$ \begin{pmatrix} 1 & 2 & 3 \ 0 & 4 & 5 \ 0 & 0 & 6 \end{pmatrix} $$