Lecture 2: Multiplying and Factoring 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

Multiplying and factoring matrices are the topics of this lecture. Professor Strang reviews multiplying columns by rows: $A B =$ sum of rank one matrices. He also introduces the five most important factorizations.

Summary

Multiply columns by rows: $A B =$ sum of rank one matrices

Five great factorizations:

$A = L U$ from elimination
$A = Q R$ from orthogonalization (Gram-Schmidt)
$S = Q Λ Q^{T}$ from eigenvectors of a symmetric matrix $S$
$A = X Λ X^{- 1}$ diagonalizes $A$ by the eigenvector matrix $X$
$A = U Σ V^{T} =$ (orthogonal)(diagonal)(orthogonal) = Singular Value Decomposition

Related section in textbook: I.2

Instructor: Prof. Gilbert Strang

Problems for Lecture 2
From textbook Section I.2

2. Suppose $a$ and $b$ are column vectors with components $a_{1}, \dots, a_{m}$ and $b_{1}, \dots, b_{p}$ . Can you multiply $a$ times $b^{T}$ (yes or no)? What is the shape of the answer $a b^{T}$ ? What number is in row $i$ , column $j$ of $a b^{T}$ ? What can you say about $a a^{T}$ ?

6. If $A$ has columns $a_{1}, a_{2}, a_{3}$ and $B = I$ is the identity matrix, what are the rank one matrices $a_{1} b_{1}^{*}$ and $a_{2} b_{2}^{*}$ and $a_{3} b_{3}^{*}$ ? They should add to $A I = A$ .