Triangular Matrix

There are two types of triangular matrices.

1. Upper Triangular Matrix: A square matrix (a_ij) is said to be an upper triangular matrix if all the elements below the principal diagonal are zero (0). That is, [a_ij]_m _{× n} is an upper triangular matrix if (i) m = n and (ii) a_ij = 0 for i > j.

Examples of an Upper Triangular Matrix are:

(i) $\begin{bmatrix} 5 & 2 & 8\\ 0 & 3 & 10\\ 0 & 0 & 8 \end{bmatrix}$

(ii) $\begin{bmatrix} -1 & 7 & 3\\ 0 & 6 & 1\\ 0 & 0 & 5 \end{bmatrix}$

(iii) $\begin{bmatrix} 3 & 0 & 3\\ 0 & 7 & -1\\ 0 & 0 & 2 \end{bmatrix}$

2. Lower Triangular Matrix: A square matrix (a_ij) is said to be a lower triangular matrix if all the elements above the principal diagonal are zero (0). That is, [a_ij]_m _{× n} is a lower triangular matrix if (i) m = n and (ii) a_ij = 0 for i < j.

Examples of a Lower Triangular Matrix are:

(i) $\begin{bmatrix} 7 & 0 & 0\\ 3 & 9 & 0\\ 1 & 2 & 1 \end{bmatrix}$

(ii) $\begin{bmatrix} 1 & 0 & 0\\ -5 & 1 & 0\\ 3 & 7 & 1 \end{bmatrix}$

(iii) $\begin{bmatrix} 9 & 0 & 0\\ 1 & 3 & 0\\ 2 & 5 & -4 \end{bmatrix}$

Definition of Triangular Matrix:

A square matrix is said to be a triangular matrix if it is either upper triangular or lower triangular.

For example:

(i) $\begin{bmatrix} 2 & 3 & 1\\ 0 & 1 & 3\\ 0 & 0 & 4 \end{bmatrix}$

(ii) $\begin{bmatrix} 1 & 0 & 0\\ 2 & 3 & 0\\ 4 & 1 & 2 \end{bmatrix}$

(iii) $\begin{bmatrix} 0 & 0 & 0\\ 3 & 0 & 0\\ 2 & 1 & 0 \end{bmatrix}$

(iv) $\begin{bmatrix} 0 & 1 & 2\\ 0 & 0 & 3\\ 0 & 0 & 0 \end{bmatrix}$

A diagonal matrix is both upper triangular and lower triangular.