Lecture 8: Norms of Vectors and Matrices

Course Info

Instructor

Prof. Gilbert Strang

Departments

Mathematics

As Taught In

Spring 2018

Level

Undergraduate

Topics

Learning Resource Types

Lecture Videos

Problem Sets

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Video Lectures

Description

A norm is a way to measure the size of a vector, a matrix, a tensor, or a function. Professor Strang reviews a variety of norms that are important to understand including S-norms, the nuclear norm, and the Frobenius norm.

Summary

The $ℓ^{1}$ and $ℓ^{2}$ and $ℓ^{\infty}$ norms of vectors
The unit ball of vectors with norm $\leq$ 1
Matrix norm = largest growth factor = max $‖ A x ‖ / ‖ x ‖$
Orthogonal matrices have $‖ Q ‖_{2} = 1$ and $‖ Q ‖_{F}^{2} = n$

Related section in textbook: I.11

Instructor: Prof. Gilbert Strang

Problems for Lecture 8
From textbook Section I.11

1. Show directly this fact about $ℓ^{1}$ and $ℓ^{2}$ and $ℓ^{\infty}$ vector norms : $| | v | |_{2}^{2} \leq | | v | |_{1} | | v | |_{\infty}$

7. A short proof of $| | A B | |_{F} \leq | | A | |_{F} | | B | |_{F}$ starts from multiplying rows times columns :
$| (A B)_{i j} |^{2} \leq | | row i of A | |^{2} | | column j of B | |^{2}$ is the Cauchy-Schwarz inequality.
Add up both sides over all $i$ and $j$ to show that $| | A B | |_{F}^{2} \leq | | A | |_{F}^{2} | | B | |_{F}^{2}$