Lower Triangular Matrix
A lower triangular matrix is a special type of square matrix where all elements above the main diagonal are zero. That is, for an \( n \times n \) matrix \( A = [a_{ij}] \), it is lower triangular if:
\[ a_{ij} = 0 \quad \text{for all } i < j \]
In other words, nonzero elements are restricted to the main diagonal and the lower part of the matrix.
Example of a Lower Triangular Matrix
Consider the following \( 3 \times 3 \) matrix:
\[ L = \begin{bmatrix} 1 & 0 & 0 \\ 4 & 2 & 0 \\ 3 & -1 & 5 \end{bmatrix} \]
All elements above the diagonal are zero, confirming that \( L \) is a lower triangular matrix.
Properties of Lower Triangular Matrices
Property 1: The Product of Two Lower Triangular Matrices is Also Lower Triangular
If \( L_1 \) and \( L_2 \) are both lower triangular \( n \times n \) matrices, then their product \( L_1 L_2 \) is also lower triangular.
Example
Consider the following lower triangular matrices:
\[ L_1 = \begin{bmatrix} 1 & 0 \\ 3 & 2 \end{bmatrix}, \quad L_2 = \begin{bmatrix} 4 & 0 \\ -1 & 5 \end{bmatrix} \]
Their product is:
\[ L_1 L_2 = \begin{bmatrix} (1 \cdot 4 + 0 \cdot -1) & (1 \cdot 0 + 0 \cdot 5) \\ (3 \cdot 4 + 2 \cdot -1) & (3 \cdot 0 + 2 \cdot 5) \end{bmatrix} = \begin{bmatrix} 4 & 0 \\ 10 & 10 \end{bmatrix} \]
This confirms that the product is still lower triangular.
Property 2: The Inverse of a Non-Singular Lower Triangular Matrix is Also Lower Triangular
If a lower triangular matrix \( L \) is non-singular (i.e., its determinant is nonzero), then its inverse \( L^{-1} \) is also lower triangular.
Example
Consider the lower triangular matrix:
\[ L = \begin{bmatrix} 1 & 0 \\ 2 & 3 \end{bmatrix} \]
The inverse is computed as:
\[ L^{-1} = \frac{1}{\det(L)} \begin{bmatrix} 3 & 0 \\ -2 & 1 \end{bmatrix} \]
Since \( L^{-1} \) is also lower triangular, this property holds.
Property 3: The Determinant of a Lower Triangular Matrix is the Product of its Diagonal Elements
If \( L \) is an \( n \times n \) lower triangular matrix, then:
\[ \det(L) = \prod_{i=1}^{n} a_{ii} \]
Example
Consider the lower triangular matrix:
\[ L = \begin{bmatrix} 2 & 0 & 0 \\ -3 & 4 & 0 \\ 1 & 5 & 6 \end{bmatrix} \]
The determinant is computed as:
\[ \det(L) = 2 \times 4 \times 6 = 48 \]
Thus, the determinant follows the stated property.
Property 4: A Lower Triangular Matrix Can Be Used for Efficient Forward Substitution
Lower triangular matrices are useful for solving linear equations using forward substitution, especially in LU decomposition.
Example
Given the system:
\[ \begin{bmatrix} 2 & 0 & 0 \\ 1 & 3 & 0 \\ 4 & -2 & 5 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 4 \\ 7 \\ 3 \end{bmatrix} \]
We solve for \( x \), \( y \), and \( z \) in a sequential manner:
- \( 2x = 4 \Rightarrow x = 2 \)
- \( 1(2) + 3y = 7 \Rightarrow y = \frac{5}{3} \)
- \( 4(2) – 2\left(\frac{5}{3}\right) + 5z = 3 \Rightarrow z = \frac{2}{5} \)
Forward substitution efficiently computes the solution.