Lower Triangular Matrix

Tags: #◇CSMATH2 #◇Linear Algebra #◇Computational Mathematics #◇Matrices #◇Matrix Operations #◇Square Matrix #◇Computer Science #◇Vector Spaces #◇Linear Transformations

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◇Overview

A lower triangular matrix is a square matrix where all entries above the main diagonal are zero.

◇Key Concepts

$a_{ij} = 0$ for all $i < j$ The main diagonal and entries below it can be nonzero Product of lower triangular matrices is lower triangular Determinant = product of diagonal entries

◇Details / Theory

Definition

A square matrix $A = [a_{ij}]$ of size $n \times n$ is lower triangular if:

$a_{ij} = 0 \quad \text{for all } i < j$

Example

$L = \begin{bmatrix} 4 & 0 & 0 \\ 2 & -3 & 0 \\ 1 & 5 & 6 \end{bmatrix}$

All entries above the main diagonal are zero.

Properties

LU Decomposition

Many numerical methods decompose $A = LU$ where $L$ is lower triangular and $U$ is upper triangular. This enables efficient solving of $A\mathbf{x} = \mathbf{b}$ via forward and back substitution.

◇Common Pitfalls

Confusing lower and upper triangular — lower has zeros above the diagonal. A matrix that is both upper and lower triangular is a ◇Diagonal Matrix.

◇Summary

TL;DR — Lower triangular matrix: all entries above the main diagonal are zero. Determinant = product of diagonal entries. Transpose of lower triangular is upper triangular.

◇Links

Related: ◇Upper Triangular Matrix — ◇Triangular Matrices — ◇Square Matrix — ◇Main Diagonal — ◇Special Types of Square Matrices — ◇Diagonal Matrix Source / Reference:

◇References & Navigation

• Previous Topic: ◇Inverse of a Matrix
• Next Topic: ◇Main Diagonal
• Related Notes: ◇Trace of a Matrix, ◇Symmetric Matrix, ◇Matrices Detailed