A square matrix in which all the elements above the leading diagonal are zero and all the entries on the main diagonal are one, is called a lower unitriangular matrix.
Let’s understand the meanings of the words “lower” and “triangular” firstly before learning the concept of the lower triangular matrix and its internal structure.
The entries above the main diagonal are zero in a square matrix in a special case. The elements on the leading diagonal and the entries below the primary diagonal form a triangle shape. Therefore, the square matrix is called a lower triangular matrix.
$L \,=\,
\begin{bmatrix}
e_{11} & 0 & 0 & \cdots & 0\\
e_{21} & e_{22} & 0 & \cdots & 0\\
e_{31} & e_{32} & e_{33} & \cdots & 0\\
\vdots & \vdots & \vdots & \ddots & \vdots\\
e_{n1} & e_{n2} & e_{n3} & \cdots & e_{nn}
\end{bmatrix}$
Basically, the matrix $L$ is a square matrix of order $n$. The elements $e_{11}$, $e_{22}$, $e_{33}$, $e_{44}$ $\cdots$ $e_{nn}$ are the entries on the major diagonal. The elements on the leading diagonal and the entries below the principal diagonal form a triangle shape but the elements above the primary diagonal are zeros. Hence, the matrix $L$ is called a lower triangular matrix.
$L \,=\,
{\begin{bmatrix}
e_{\displaystyle ij}
\end{bmatrix}}_{\displaystyle n \times n}$
There is a mathematical condition in the case of a lower triangular matrix. The entries are zero if $i < j$ and they are elements above the leading diagonal of the matrix. The remaining entries are non-zeros and form a triangle shape.
It means $e_{ij} = 0$, if $i < j$.
Observe the below three matrices to know the concept of lower triangular matrix.
$A \,=\,
\begin{bmatrix}
1 & 0\\
6 & -5
\end{bmatrix}$
The matrix $A$ is a second order square matrix.
Hence, the matrix $A$ is called a lower triangular matrix.
$B \,=\,
\begin{bmatrix}
9 & 0 & 0\\
4 & 1 & 0\\
7 & 2 & -3
\end{bmatrix}$
The matrix $B$ is a third order square matrix.
So, the matrix $B$ is called a lower triangular matrix.
$C =
\begin{bmatrix}
5 & 0 & 0 & 0\\
-1 & 6 & 0 & 0\\
8 & 3 & 9 & 0\\
-2 & -6 & -7 & 2\\
\end{bmatrix}$
The matrix $C$ is a fourth order square matrix.
Therefore, the matrix $C$ is called a lower triangular matrix.
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