Scalar Multiplication

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

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◇Overview

In matrix contexts, real numbers are often referred to as scalars. (Handout 1, Page 9)

◇Details / Theory

In matrix contexts, real numbers are often referred to as scalars. (Handout 1, Page 9) Definition: If $A=[a_{ij}]$ is an $m \times n$ matrix and $c$ is a scalar, the scalar multiple $cA$ is the $m \times n$ matrix obtained by multiplying each entry of $A$ by $c$:
$cA=[c \cdot a_{ij}]$
(Handout 1, Page 9) Example:
$3\begin{bmatrix} 1 & 2 & 3 \\ 4 & -1 & 2 \\ 0 & 3 & -4 \end{bmatrix}=\begin{bmatrix} 3(1) & 3(2) & 3(3) \\ 3(4) & 3(-1) & 3(2) \\ 3(0) & 3(3) & 3(-4) \end{bmatrix}=\begin{bmatrix} 3 & 6 & 9 \\ 12 & -3 & 6 \\ 0 & 9 & -12 \end{bmatrix}$
(Handout 1, Page 9)
PN: Scalar multiplication scales the entire matrix. Every element gets multiplied by the scalar.

◇Common Pitfalls

Scalar multiplication multiplies every entry: $(kA)_{ij} = k \cdot a_{ij}$. Don’t confuse with matrix multiplication — scalar multiplication always commutes: $kA = Ak$.

◇Summary

TL;DR — $kA$ multiplies every entry of $A$ by scalar $k$. Distributes over addition and is associative: $k(lA) = (kl)A$.

◇Links

Related: Source / Reference:

◇References & Navigation

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• Related Notes: ◇Symmetric Matrix, ◇Upper Triangular Matrix