Linear and multilinear algebra; matrix theory Definition

Triangular matrix

Builds on Ring Square matrix Upper triangular matrix Lower triangular matrix

Definition

Let $R$ be a ring and let $A = (a_{ij})$ be an $n \times n$ square matrix over $R$ . The matrix $A$ is upper triangular when $a_{ij} = 0$ for all $i > j$ . It is lower triangular when $a_{ij} = 0$ for all $i < j$ . It is triangular when it is upper triangular or lower triangular.

Remarks

Example: $\begin{pmatrix} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 0 & 0 & 6 \end{pmatrix}$ is triangular, because its entries below the diagonal, in positions $(2,1)$ , $(3,1)$ , $(3,2)$ , are zero. Non-example: $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ is not triangular, because $a_{21} = 3 \ne 0$ and $a_{12} = 2 \ne 0$ . A matrix is both upper and lower triangular exactly when it is diagonal. A triangular matrix is strictly triangular when its diagonal entries are also zero, and unit triangular when they are all $1$ . See also upper triangular matrix and lower triangular matrix.

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Tag	`3AQFY4PAC2`
URI	`m2://3AQFY4PAC2`
Kind	definition
Writing	W0 stub
MSC	15B99, 15A23
Defines	Triangular matrix
Uses	Ring, Square matrix, Upper triangular matrix, Lower triangular matrix
Changed	2 November 2026

Triangular matrix

Definition

Remarks

Reading path

Further reading

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