A square matrix in which all the entries below the main diagonal are zero, is called an upper triangular matrix.
Firstly, let’s know the meanings of the words “upper” and “triangular” before learning the concept of an upper triangular matrix and its internal structure.
In a special case, the elements below the leading diagonal are zero in a square matrix, the entries on the main diagonal and elements above the principal diagonal form a triangle shape. Hence, the square matrix is called an upper triangular matrix.
$U \,=\, \begin{bmatrix} e_{11} & e_{12} & e_{13} & \cdots & e_{1n}\\ 0 & e_{22} & e_{23} & \cdots & e_{2n}\\ 0 & 0 & e_{33} & \cdots & e_{3n}\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & 0 & \cdots & e_{nn} \end{bmatrix} $
The matrix $U$ is basically a square matrix of order $n$. In this matrix, $e_{11}$, $e_{22}$, $e_{33}$, $e_{44}$ $\cdots$ $e_{nn}$ are the elements on the primary diagonal. The entries on the major diagonal and the elements above the main diagonal form a triangle shape but the entries below the leading diagonal are zeros. Therefore, the matrix $U$ is called an upper triangular matrix.
$U \,=\, {\begin{bmatrix} e_{\displaystyle ij} \end{bmatrix}}_{\displaystyle n \times n}$
In an upper triangular matrix, the elements are zero if $i > j$ and they are elements under the main diagonal of the matrix but the remaining elements are non-zero elements and form a triangle shape.
In other words, $e_{ij} = 0$ if $i > j$.
Look at the following three matrices to understand the concept of upper triangular matrix.
$A \,=\, \begin{bmatrix} 2 & -4\\ 0 & 3 \end{bmatrix}$
The matrix $A$ is a square matrix of order $2$.
So, the matrix $A$ is called an upper triangular matrix.
$B \,=\, \begin{bmatrix} 1 & 5 & 6\\ 0 & 3 & 7\\ 0 & 0 & 4 \end{bmatrix}$
The matrix $B$ is a square matrix of order $3$.
Hence, the matrix $B$ is called an upper triangular matrix.
$C = \begin{bmatrix} 4 & 2 & -3 & 9\\ 0 & -1 & 6 & -5\\ 0 & 0 & 8 & 3\\ 0 & 0 & 0 & 5\\ \end{bmatrix}$
The matrix $C$ is a square matrix of the order $4$.
Therefore, the matrix $C$ is called an upper triangular matrix.
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