Lower Triangular Matrix/Examples

Examples of Lower Triangular Matrices

Lower Triangular Matrix with fewer Rows than Columns

A lower triangular matrix of order $m \times n$ such that $m < n$:

$\mathbf L = \begin{bmatrix}

    a_{1 1} &            0 &            0 & \cdots &                0 &       0 & \cdots &      0 &     0 \\
    a_{2 1} &      a_{2 2} &            0 & \cdots &                0 &       0 & \cdots &      0 &     0 \\
    a_{3 1} &      a_{3 2} &      a_{3 3} & \cdots &                0 &       0 & \cdots &      0 &     0 \\
     \vdots &       \vdots &       \vdots & \ddots &           \vdots &  \vdots & \ddots & \vdots & \vdots \\

a_{m - 1, 1} & a_{m - 1, 2} & a_{m - 1, 3} & \cdots & a_{m - 1, m - 1} & 0 & \cdots & 0 & 0 \\

    a_{m 1} &      a_{m 2} &      a_{m 3} & \cdots &     a_{m - 1, m} & a_{m m} & \cdots &      0 &     0 \\

\end{bmatrix}$

Lower Triangular Matrix with more Rows than Columns

A lower triangular matrix of order $m \times n$ such that $m > n$:

$\mathbf L = \begin{bmatrix}

    a_{1 1} &            0 &            0 & \cdots &                0 &            0 \\
    a_{2 1} &      a_{2 2} &            0 & \cdots &                0 &            0 \\
    a_{3 1} &      a_{3 2} &      a_{3 3} & \cdots &                0 &            0 \\
     \vdots &       \vdots &       \vdots & \ddots &           \vdots &       \vdots \\

a_{n - 1, 1} & a_{n - 1, 2} & a_{n - 1, 3} & \cdots & a_{n - 1, n - 1} & 0 \\

    a_{n 1} &      a_{n 2} &      a_{n 3} & \cdots &     a_{n, n - 1} &      a_{n n} \\

a_{n + 1, 1} & a_{n + 1, 2} & a_{n + 1, 3} & \cdots & a_{n + 1, n - 1} & a_{n + 1, n} \\

     \vdots &       \vdots &       \vdots & \ddots &           \vdots &       \vdots \\

a_{m - 1, 1} & a_{m - 1, 2} & a_{m - 1, 3} & \cdots & a_{m - 1, n - 1} & a_{m - 1, n} \\

    a_{m 1} &      a_{m 2} &      a_{m 3} & \cdots &     a_{m, n - 1} &      a_{m n} \\

\end{bmatrix}$

Square Lower Triangular Matrix

An lower triangular square matrix of order $n$:

$\mathbf L = \begin{bmatrix}

    a_{1 1} &            0 &            0 & \cdots &                0 &            0 \\
    a_{2 1} &      a_{2 2} &            0 & \cdots &                0 &            0 \\
    a_{3 1} &      a_{3 2} &      a_{3 3} & \cdots &                0 &            0 \\
     \vdots &       \vdots &       \vdots & \ddots &           \vdots &       \vdots \\

a_{n - 1, 1} & a_{n - 1, 2} & a_{n - 1, 3} & \cdots & a_{n - 1, n - 1} & 0 \\

    a_{n 1} &      a_{n 2} &      a_{n 3} & \cdots &     a_{n, n - 1} &      a_{n n} \\

\end{bmatrix}$