Upper Triangular Matrix

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

◇Overview

An upper triangular matrix is a square matrix where all entries below the main diagonal are zero.

◇Key Concepts

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

◇Details / Theory

Definition

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

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

Example

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

All entries below the main diagonal are zero.

Properties

Property	Result
Product of upper triangular	Upper triangular
Sum of upper triangular	Upper triangular
Inverse (if exists)	Upper triangular
Determinant	$\det(U) = u_{11} \cdot u_{22} \cdots u_{nn}$
Transpose	◇Lower Triangular Matrix

Relationship to Row Echelon Form

Gaussian elimination transforms a matrix into an upper triangular form (row echelon form), which simplifies solving $A\mathbf{x} = \mathbf{b}$ via back substitution.

◇Common Pitfalls

Confusing upper and lower triangular — upper has zeros below the diagonal. Diagonal entries can be zero (but then the matrix is singular).

◇Summary

TL;DR — Upper triangular matrix: all entries below the main diagonal are zero. Its determinant is the product of diagonal entries. Transpose of upper triangular is lower triangular.

◇Links

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

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Related Notes: ◇Main Diagonal, ◇Definition of a Matrix, ◇Matrices Detailed

UPPER TRIANGULAR MATRIX