Matrix Multiplication

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

◇Overview

Definition: If $A=[a_{ij}]$ is an $m \times n$ matrix and $B=[b_{ij}]$ is an $n \times p$ matrix, their product $AB$ is an $m \times p$ matrix $C=[c_{ij}]$

◇Key Concepts

Size Compatibility Condition Non-Commutativity

◇Details / Theory

Definition: If $A=[a_{ij}]$ is an $m \times n$ matrix and $B=[b_{ij}]$ is an $n \times p$ matrix, their product $AB$ is an $m \times p$ matrix $C=[c_{ij}]$, where the entry $c_{ij}$ (in the i-th row, j-th column of $C$) is calculated by taking the dot product of the i-th row of $A$ with the j-th column of $B$:
$c_{ij}=(\text{i-th row of } A) \cdot (\text{j-th column of } B)=\sum_{k=1}^{n}a_{ik}b_{kj}$
$c_{ij}=a_{i1}b_{1j}+a_{i2}b_{2j}+\cdots+a_{in}b_{nj}$
(Handout 1, Page 21) Size Compatibility Condition: The number of columns of $A$ (n) must equal the number of rows of $B$ (n). If $A$ is $m \times n$ and $B$ is $n \times p$, then $AB$ is defined and has size $m \times p$. (Handout 1, Page 21)
PN: This is the most complex standard operation. It's NOT element-wise multiplication! Always check dimensions first: $(m \times n) \times (n \times p) \rightarrow (m \times p)$. The inner dimensions $n$ must match. Non-Commutativity: Matrix multiplication is generally NOT commutative, meaning $AB \neq BA$ even if both products are defined (which requires $A$ and $B$ to be square matrices of the same size). (Handout 1, Page 25)
Example: Let $A=\begin{bmatrix} 3 & -2 \\ 2 & 5 \end{bmatrix}$ and $B=\begin{bmatrix} 0 & 1 \\ 3 & 4 \end{bmatrix}$.
$AB=\begin{bmatrix} 3(0)+(-2)(3) & 3(1)+(-2)(4) \\ 2(0)+5(3) & 2(1)+5(4) \end{bmatrix}=\begin{bmatrix} -6 & -5 \\ 15 & 22 \end{bmatrix}$

    $$BA=\begin{bmatrix} 0(3)+1(2) & 0(-2)+1(5) \\ 3(3)+4(2) & 3(-2)+4(5) \end{bmatrix}=\begin{bmatrix} 2 & 5 \\ 17 & 14 \end{bmatrix}$$

    Clearly, $AB \neq BA$. (Handout 1, Page 25)

PN: This is a critical difference from scalar arithmetic. The order of multiplication matters significantly.

◇Common Pitfalls

Multiplication is NOT commutative: $AB \neq BA$ in general. Inner dimensions must match: $A_{m \times n} \cdot B_{n \times p}$ — if cols of $A \neq$ rows of $B$, undefined. $AB = O$ does not mean $A = O$ or $B = O$.

◇Summary

TL;DR — $(AB)_{ij} = \sum_k a_{ik} b_{kj}$. Requires inner dimensions to match. Not commutative. Associative and distributive over addition.

◇Links

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◇References & Navigation

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• Related Notes: ◇Upper Triangular Matrix, ◇Types of Matrices, ◇Matrices Detailed