Scalar Matrix

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

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◇Overview

Definition: A Diagonal Matrix whose main diagonal entries are all equal is called a scalar matrix. (Handout 2, Page 4)

◇Key Concepts

Definition

◇Details / Theory

Definition: A ◇Diagonal Matrix whose main diagonal entries are all equal is called a scalar matrix. (Handout 2, Page 4)
So, aij​=0 for i=j, and aii​=k for some constant k. Example:
S=[−2​0​0 0​−2​0 0​0​−2​]
(Handout 2, Page 4, modified example)
A scalar matrix can be written as kIn​, where In​ is the identity matrix.

◇Common Pitfalls

A scalar matrix $kI$ commutes with every matrix of the same size: $kI \cdot A = A \cdot kI = kA$. Don’t confuse scalar multiplication ($kA$) with the scalar matrix ($kI$) — they are related but different concepts.

◇Summary

TL;DR — A scalar matrix is $kI$ — a diagonal matrix with identical diagonal entries. It acts like scalar multiplication when multiplied with any matrix: $kI \cdot A = kA$.

◇Links

Related: - ◇Diagonal Matrix Source / Reference:

◇References & Navigation

• Previous Topic: ◇Row Matrix
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• Related Notes: ◇Types of Matrices, ◇Powers of a Matrix, ◇Matrices Detailed