# Category:Triangular Matrices

This category contains results about **Triangular Matrices**.

Definitions specific to this category can be found in Definitions/Triangular Matrices.

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:

- $\forall a_{i j} \in \mathbf U: i < j \implies a_{i j} = 0$

## Subcategories

This category has the following 3 subcategories, out of 3 total.

## Pages in category "Triangular Matrices"

The following 10 pages are in this category, out of 10 total.