Matrix Subtraction (Difference)

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

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◇Overview

Definition: If $A=[a_{ij}]$ and $B=[b_{ij}]$ are matrices of the same size ($m \times n$), their difference $A-B$ is defined as $A+(-1)B$. This res...

◇Key Concepts

Non-Commutativity

◇Details / Theory

Definition: If $A=[a_{ij}]$ and $B=[b_{ij}]$ are matrices of the same size ($m \times n$), their difference $A-B$ is defined as $A+(-1)B$. This results in subtracting corresponding entries:
$A-B=A+(-1)B=[a_{ij}]+[-b_{ij}]=[a_{ij}-b_{ij}]$
(Handout 1, Pages 10) Example: Let $A=\begin{bmatrix} 2 & 3 & 4 \\ 0 & 1 & -1 \\ 2 & 0 & 1 \end{bmatrix}$ and $B=\begin{bmatrix} 0 & 6 & 2 \\ 4 & 1 & 0 \\ -1 & 2 & 4 \end{bmatrix}$.
$A-B=\begin{bmatrix} 2-0 & 3-6 & 4-2 \\ 0-4 & 1-1 & -1-0 \\ 2-(-1) & 0-2 & 1-4 \end{bmatrix}=\begin{bmatrix} 2 & -3 & 2 \\ -4 & 0 & -1 \\ 3 & -2 & -3 \end{bmatrix}$
(Handout 1, Page 11) Non-Commutativity: In general, $A-B \neq B-A$. (Handout 1, Page 13)

◇Common Pitfalls

Same size requirement as addition: $A - B$ only defined when $A$ and $B$ are the same $m \times n$. $A - B \neq B - A$ in general (subtraction is not commutative).

◇Summary

TL;DR — Matrix subtraction: $(A - B)_{ij} = a_{ij} - b_{ij}$. Requires same dimensions. Not commutative.

◇Links

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◇References & Navigation

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