Definition:Triangular Matrix
Definition
Let $\mathbf T = \begin {bmatrix} a_{1 1} & a_{1 2} & \cdots & a_{1 n} \\ a_{2 1} & a_{2 2} & \cdots & a_{2 n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m 1} & a_{m 2} & \cdots & a_{m n} \\ \end {bmatrix}$ be a matrix of order $m \times n$.
Then $\mathbf T$ is a triangular matrix if and only if all the elements either above or below the diagonal are zero.
Upper Triangular Matrix
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.
That is, $\mathbf U$ is upper triangular if and only if:
- $\forall a_{ij} \in \mathbf U: i > j \implies a_{ij} = 0$
Lower Triangular Matrix
A lower triangular matrix is a matrix in which all the upper triangular elements are zero.
That is, all the non-zero elements are on the main diagonal or in the lower triangle.
That is, $\mathbf L$ is lower triangular if and only if:
- $\forall a_{i j} \in \mathbf U: i < j \implies a_{i j} = 0$
Also defined as
Some sources define a triangular matrix only as a square matrix.
Also see
Sources
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): triangular matrix