Diagonal Matrix

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

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◇Overview

Definition: An n×n matrix A=[aij​] is called a diagonal matrix if all its non-diagonal entries are zero, i.e., aij​=0 for i=j. (Handout 2, Page 4)

◇Key Concepts

Definition

◇Details / Theory

Definition: An n×n matrix A=[aij​] is called a diagonal matrix if all its non-diagonal entries are zero, i.e., aij​=0 for i=j. (Handout 2, Page 4) Example:
D=[1​0​0 0​−5​0 0​0​3​]
(Handout 2, Page 4, modified example)
Personal Note: Only the main diagonal entries can be non-zero, but they can also be zero.

◇Common Pitfalls

A diagonal matrix can have zeros on the diagonal — "diagonal" means off-diagonal entries are zero, not that diagonal entries are nonzero. Diagonal matrices are a special case of both upper and lower triangular.

◇Summary

TL;DR — A diagonal matrix is a square matrix where $a_{ij} = 0$ for $i \neq j$. Only the main diagonal can have nonzero entries. Diagonal matrices commute and are easy to compute with.

◇Links

Related: Source / Reference:

◇References & Navigation

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• Related Notes: ◇Triangular Matrices, ◇Lower Triangular Matrix, ◇Matrices Detailed