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.
From Triangular Matrix to HOME
Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.
Apr 13, 24 05:12 PM
Apr 13, 24 03:29 PM
Apr 13, 24 01:27 PM
Apr 13, 24 12:41 PM
Apr 12, 24 04:22 PM
New! Comments
Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.