📂Matrix Algebra

Triangular Matrix

Definition¹

A matrix $A = [a_{ij}]$ with all elements above the main diagonal being $0$ is called a lower triangular matrix.

$$A \text{ is lower triangluar matrix if } a_{ij} = 0 \text{ whenever } i \lt j$$

A matrix $A = [a_{ij}]$ with all elements below the main diagonal being $0$ is called an upper triangular matrix.

$$A \text{ is upper triangluar matrix if } a_{ij} = 0 \text{ whenever } i \gt j$$

Especially, a triangular matrix with all main diagonal elements being $0$ is called a strictly (upper/lower) triangular matrix.

Description

For example, let’s say $A$ is $4 \times 5$. If $A$ is a lower triangular matrix, then

$$A= \begin{bmatrix} a_{11} & 0 & 0 & 0 & 0 \\ a_{21} & a_{22} & 0 & 0 & 0 \\ a_{31} & a_{32} & a_{33} & 0 & 0 \\ a_{41} & a_{42} & a_{43} & a_{44} & 0 \\ \end{bmatrix}$$

If it is an upper triangular matrix, then it is as follows.

$$A= \begin{bmatrix} a_{11} & a_{12} & a_{13} & a_{14} & a_{15} \\ 0 & a_{22} & a_{23} & a_{24} & a_{25} \\ 0 & 0 & a_{33} & a_{34} & a_{35} \\ 0 & 0 & 0 & a_{44} & a_{44} \\ \end{bmatrix}$$

According to the definition, a diagonal matrix is both a lower triangular matrix and an upper triangular matrix.

Properties

• The transpose of a lower triangular matrix is an upper triangular matrix, and the transpose of an upper triangular matrix is a lower triangular matrix.

• The product of lower triangular matrices is a lower triangular matrix, and the product of upper triangular matrices is an upper triangular matrix.

• A necessary and sufficient condition for a triangular matrix to be invertible is that all main diagonal elements are not $0$.

• The inverse of an invertible lower triangular matrix is a lower triangular matrix, and the inverse of an invertible upper triangular matrix is an upper triangular matrix.

• A strictly triangular square matrix is nilpotent. (The converse is not true)