Upper triangular matrix

This article defines a property that can be evaluated for a square matrix. In other words, given a square matrix (a matrix with an equal number of rows and columns) the matrix either satisfies or does not satisfy the property.
View other properties of square matrices

Definition

Verbal definition

A square matrix is termed an upper triangular matrix if all its entries below the main diagonal are zero.

Algebraic description

Suppose $n$ is a positive integer. A $n\times n$ square matrix $A=(a_{ij})_{1\leq i\leq n,1\leq j\leq n}$ is termed an upper triangular matrix if the following holds:

$a_{ij}=\ \forall \ i>j\in \{1,2,\dots ,n\}$

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
scalar matrix	scalar multiple of identity matrix	(via diagonal matrix)	(via diagonal matrix)	Diagonal matrix|FULL LIST, MORE INFO
diagonal matrix	all entries not on the main diagonal are zero		the matrix Failed to parse (unknown function "\begin{pmatrix}"): {\displaystyle \begin{pmatrix} 1 & 1 \\ 0 & 1 \\\\end{pmatrix}} is a counterexample	|FULL LIST, MORE INFO
upper unitriangular matrix	upper triangular matrix with all diagonal entries equal to 1			|FULL LIST, MORE INFO
strictly upper triangular matrix	all entries on and below the main diagonal are zero			|FULL LIST, MORE INFO

Contrast properties=

• Lower triangular matrix: A matrix is upper triangular and lower triangular if and only if it is a diagonal matrix.
• Orthogonal matrix