Matrix Addition

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 sum $A+B$ is the $m \times n$ matrix obtained by a...

◇Key Concepts

Requirement

◇Details / Theory

Definition: If $A=[a_{ij}]$ and $B=[b_{ij}]$ are matrices of the same size ($m \times n$), their sum $A+B$ is the $m \times n$ matrix obtained by adding corresponding entries:
$A+B=[a_{ij}+b_{ij}]$
(Handout 1, Page 8) Requirement: Matrices must have the same dimensions. Addition is undefined otherwise. (Handout 1, Page 8)
PN: Conceptually simple: just add element-wise, like adding vectors. The size constraint is key. Example:
$\begin{bmatrix} -1 & 2 \\ 0 & 1 \end{bmatrix}+\begin{bmatrix} 1 & 3 \\ -1 & 2 \end{bmatrix}=\begin{bmatrix} -1+1 & 2+3 \\ 0+(-1) & 1+2 \end{bmatrix}=\begin{bmatrix} 0 & 5 \\ -1 & 3 \end{bmatrix}$
(Handout 1, Page 8)

◇Common Pitfalls

Cannot add matrices of different sizes — $A + B$ requires $A$ and $B$ to be the same $m \times n$. Addition is entry-wise, not row-wise or column-wise as a whole.

◇Summary

TL;DR — Matrix addition: $(A + B)_{ij} = a_{ij} + b_{ij}$. Requires same dimensions. Commutative and associative.

◇Links

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◇References & Navigation

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• Related Notes: ◇Scalar Multiplication, ◇Inverse of a Matrix, ◇Matrices Detailed