Definition:Triangular Matrix/Lower Triangular Matrix

Definition

A lower triangular matrix is a matrix in which all the upper triangular elements are zero.

That is, all the non-zero elements are on the main diagonal or in the lower triangle.

That is, $\mathbf L$ is lower triangular if and only if:

$\forall a_{i j} \in \mathbf U: i < j \implies a_{i j} = 0$

Also defined as

Some sources define a lower triangular matrix only as a square matrix.

Examples

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}$

Also see

• Definition:Lower Triangular Element

• Definition:Upper Triangular Matrix

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): lower triangular matrix
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): triangular matrix
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): lower triangular matrix
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): triangular matrix
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): lower triangular matrix
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): triangular matrix