# Positive semidefinite matrices.

Friday, Nov 30, 2012

7.6.1, 7.6.7

# Low-rank approximation.

Thursday, Nov 29, 2012

# PCA and Low-rank approximation

Wednesday, Nov 28, 2012

in the notes

# Principal Components Analysis

Monday, Nov 26, 2012
We will look at PCA, which is a method for analyzing datasets. The mathematics of PCA leads to the ideas of low-rank approximation and positive matrices.

# Properties of the SVD

Monday, Nov 19, 2012

in the notes

# SVD

Friday, Nov 16, 2012
We prove SVD.

## Problems

5.12.1, 5.12.2, and problems in the notes

# Schur and Spectral theorem

Thursday, Nov 15, 2012
We prove Schur triangularization and Spectral theorem.

## Problems

7.5.1, 7.5.2, 7.5.3, 7.5.4, 7.5.8, 7.5.10, 7.5.13

# Factorizations: review, SVD, spectral, Schur.

Wednesday, Nov 14, 2012
Today we discuss without proof three factroizations that play an important role in linear algebra: spectral decomposition, Schur triangulaization, and singular value decomposition (SVD). We make a few observations about these factorizations.

## Problems

Verify Cayley-Hamilton for triangular matrices

# Test 3

Friday, Nov 09, 2012

# Review day

Thursday, Nov 08, 2012

# Diagonalizable matrices

Wednesday, Nov 07, 2012

# Algebraic and Geometric multiplicities.

Monday, Nov 05, 2012

## Problems

7.2.1, 7.2.2, 7.2.3, 7.2.4, 7.2.5, 7.2.9, 7.2.12, 7.2.17, 7.2.21

# Eigenvalues, II

Friday, Nov 02, 2012

## Problems

7.1.5, 7.1.8, 7.1.9, 7.1.18

# Eigenvalues, I

Thursday, Nov 01, 2012

## Problems

7.1.1, 7.1.3, 7.1.4

# Least-squares

Wednesday, Oct 31, 2012
A discussion of the least-squares method for fitting functions to data. This topic is covered in 4.6 and 5.14.

4.6.7, 4.6.9

# Quiz and problems

Friday, Oct 26, 2012

# Orthogonal projections

Thursday, Oct 25, 2012
Finishing up projections

## Problems

5.13.1, 5.13.2, 5.13.3, 5.13.5, 5.13.6, , 5.13.11, 5.13.12, 5.13.13

# The URV Factorization and projections

Wednesday, Oct 24, 2012
We prove the URV factorization. We will then move on to 5.13 and discuss orthogonal projections

## Problems

5.13.1, 5.13.2, 5.13.3, 5.13.5, 5.13.6, , 5.13.11, 5.13.12, 5.13.13

# Orthogonal Complements

Monday, Oct 22, 2012
We define orthogonal complements and examine some properties.

## Problems

5.11.1, 5.11.3, 5.11.4, 5.11.5, 5.11.6, 5.11.8, 5.11.11, 5.11.13

# Test 2

Friday, Oct 19, 2012

# Test 2 review

Thursday, Oct 18, 2012

# Projections and idempotents

Wednesday, Oct 17, 2012
We look at the connection between projections and idempotents. In particular we show that these two classes of linear transformations are the same and that the range and null space of an idempotent are a pair of complementary subspaces.

# Complementary Subspaces

Monday, Oct 15, 2012
Complementary subspaces will play an important role in the development of linear algebra from this point forward. Especially important is the fact that a pair of complemenary subspaces gives rise to a projection

## Problems

5.9.1, 5.9.3, 5.9.4, 5.9.5, 5.9.6, 5.9.8

# Discrete Fourier Transform

Friday, Oct 12, 2012

## Problems

5.8.1, 5.8.2, 5.8.3, 5.8.5, 5.8.10

# Householder reduction

Thursday, Oct 11, 2012

## Problems

5.7.1, 5.7.2, 5.7.3

# Elementary reflectors and projectors

Wednesday, Oct 10, 2012

## Problems

5.7.1, 5.7.2, 5.7.3

# Orthogonal and unitary matrices

Friday, Oct 05, 2012

## Problems

5.6.1(b)&(c), 5.6.2, 5.6.3, 5.6.5(a)&(b), 5.6.8(a), 5.6.10, 5.6.13

# QR factorization

Thursday, Oct 04, 2012

## Problems

5.5.6, 5.5.8, 5.5.11

# Parallelogram law, quiz discussion, Gram-Schmidt wrap-up

Wednesday, Oct 03, 2012

# Gram-Schmidt Orthonormalization

Monday, Oct 01, 2012

## Problems

5.5.1, 5.5.2, 5.5.3, 5.5.5

# Orthogonal vectors

Friday, Sep 28, 2012

## Problems

5.4.1(b)&(c), 5.4.3, 5.4.4, 5.4.6, 5.4.7, 5.4.8, 5.4.9, 5.4.16

# Inner product spaces

Thursday, Sep 27, 2012

## Problems

5.3.1, 5.3.2, 5.3.3, 5.3.4, 5.3.5

# Matrix norms

Wednesday, Sep 26, 2012

## Problems

5.2.1, 5.2.2, 5.2.3, 5.2.4, 5.2.5

## Reminders

• Read chapter 5.3 for Thursday

# Matrix norms

Monday, Sep 24, 2012

## Problems

5.2.1, 5.2.2, 5.2.3, 5.2.4, 5.2.5

## Reminders

• Read chapter 5.3 for Wednesday

# Lagrange Multipliers

Friday, Sep 21, 2012

# Vector norms

Thursday, Sep 20, 2012

## Problems

5.1.1, 5.1.2 , 5.1.3 , 5.1.4 , 5.1.5 , 5.1.6 , 5.1.8

## Reminders

• Read chapter 5.2 for Friday
• Look up Lagrange multipliers

# Vector norms

Wednesday, Sep 19, 2012

## Problems

5.1.1, 5.1.2 , 5.1.3 , 5.1.4 , 5.1.5 , 5.1.6 , 5.1.8

## Reminders

• Read chapter 5.2 for Thursday
• Try checking that the three functions defined at the end of class are norms

# Change of basis and similarity

Monday, Sep 17, 2012

## Problems

4.8.1 , 4.8.2 , 4.8.3 , 4.8.6 , 4.8.8

## Reminders

• Read chapter 5.1 for Wednesday

# Exam 1

Friday, Sep 14, 2012

# Review day

Thursday, Sep 13, 2012
Quiz 2 explanation. Followed by Q&A. 4.8 not on exam 1

# Change of basis and Similarity

Wednesday, Sep 12, 2012
We tried to figure out what happens to the matrix of a linear transformation when we change bases. Then we tried to see what happens to a vector under a change of basis. You now have four homework problems that relate to this topic.

## Problems

4.8.1 , 4.8.2 , 4.8.3 , 4.8.6 , 4.8.8

## Reminders

• Work through all the problems that have been assigned. We have a test and quiz coming up.

# Matrix of a linear transformation

Monday, Sep 10, 2012

# Linear Transformations, part 2

Friday, Sep 07, 2012
Coordinates, the matrix of linear transformation with respect to a basis.

## Problems

4.7.11 , 4.7.12 , 4.7.14 , 4.7.17

# Linear Transformations, part 1

Thursday, Sep 06, 2012
Definition and the matrix of a linear transformation

# Problem session

Wednesday, Sep 05, 2012
Work problems from 4.4.

# Basis and Dimension

Tuesday, Sep 04, 2012
Definition of a basis, dimensions of the four fundamental subspaces, computing a basis for each of these.

## Problems

4.4.2 , 4.4.3 , 4.4.4 , 4.4.6 , 4.4.7 , 4.4.8 , 4.4.17 , 4.4.18

## Reminders

• Read chapter 4.5 for Wednesday

# Linear independence

Friday, Aug 31, 2012
Computing spanning sets for the range and kernel using row reduction. Definition of linear independence and basis.

## Problems

4.3.1(a)&(c) , 4.3.5 , 4.3.7 , 4.3.8 , 4.3.10 , 4.3.12 , 4.3.13(b)&(c)

## Reminders

• Read chapter 4.4 for Tuesday

# The four fundamental subspaces

Thursday, Aug 30, 2012
The relationship between the range, kernel, orthogonal complement of a matrix and its transpose.

## Problems

4.2.1 , 4.2.2 , 4.2.3 , 4.2.5 , 4.2.8 , 4.2.10 , 4.2.12

## Reminders

• Read chapter 4.3 for Friday

# Vector spaces, part II

Wednesday, Aug 29, 2012
Definition of a vector space and some examples. Spanning sets.

## Problems

4.1.7, 4.1.8, 4.1.9, 4.1.11

## Reminders

• Read chapter 4.2, 4.3 for Thusday and Friday

# Vector spaces

Monday, Aug 27, 2012
Review that R^n is a vector space and show define a subspace. Somce examples of subspaces.

## Problems

4.1.1, 4.1.2, 4.1.5, 4.1.6

## Reminders

• Read chapter 4.1 for Wednesday

# LU Decomposition

Friday, Aug 24, 2012
Statement and proof of the LU decomposition. Some examples.

## Problems

3.10.1, 3.10.2, 3.10.3, 3.10.6, 3.10.9

## Reminders

• Read chapter 4.1 for Monday

# Elementary matrices

Thursday, Aug 23, 2012
Elementary matrices and their connection to row reduction. Proof that every non-singular matrix is the product of elementary matrices.

## Problems

3.9.1, 3.9.3, 3.9.4, 3.9.5, 3.9.7, 3.9.8, 3.9.9

## Reminders

• Read chapter 3.10 for Friday

# Matrix Inversion

Wednesday, Aug 22, 2012
Conditions for a matrix to be invertible.

## Problems

3.7.1(d) and (e), 3.7.3, 3.7.4, 3.7.6, 3.7.8, 3.7.9, 3.7.11(a)

## Reminders

• Read chapter 3.9 for Thursday

# Review of linear systems

Monday, Aug 20, 2012
Gaussian elimination, row echelon form, rank.

## Problems

1.2.1, 2.1.1(b), 2.1.3, 2.1.6, 2.2.1(a), 2.3.1(c), 2.3.3, 2.4.7, 2.5.1(a), 2.5.4

## Reminders

• Read chapter 3.7 for Wednesday.

# Course Overview

## Instructor

Mrinal Raghupathi

This class is a continuation of SM261. A central theme in linear algebra and matrix analysis is the notion of a matrix factorization. These factorizations have important applications in a wide variety of applications. In this class we look at the QR decompositions, SVD and the spectral theorem. We will develop the linear algebraic machinery needed to appreciate these results

Course Policy Statement

Syllabus

## Textbook

Matrix Analysis and Applied Linear Algebra, by Carl D. Meyer. First edition, SIAM.

# Quizzes

• Thu, Aug 30, 2012
• Topic: 1.2, 2.1 — 2.5; 3.7, 3.9, 3.10
• Thu, Sep 06, 2012
• Topic: 4.1, 4.2, 4.3
• Mon, Sep 24, 2012
• Topic: 4.8, 5.1
• Mon, Oct 01, 2012
• Topic: 5.2, 5.3
• Thu, Oct 11, 2012
• Topic: 5.4, 5.5, 5.6
• Fri, Oct 26, 2012
• Topic: 5.9, 5.11, 5.13
• Wed, Nov 21, 2012
• Topic: Normal matrices, SVD, Spectral theorem, Schur Triangularization (chapter 7.5, 5.12 and notes)

# Exams

• Exam 1
• Friday, Sep 14, 2012
• 1.2, 2.1 — 2.5; 3.7, 3.9, 3.10, 4.1 — 4.5, 4.7.
• Review 9/13, in class.
• Exam 2
• Friday, Oct 19, 2012
• 4.8, 5.1 — 5.9.
• Review 10/17, during evening EI 2000 -- 2100
• Exam 3
• Friday, Nov 09, 2012
• 5.11, 5.13, 4.6, 7.1 — 7.2.
• Review 11/08, during class.
• Exam 4
• Monday, Dec 03, 2012
• TBA
• TBA

# Notes

• Schur triangularization, spectral theorem, and SVD [PDF]
• Principal Components Analysis [PDF]
• Low-rank approximation and perturbation of singular values. [PDF]

# Quotes

''We share a philosophy about linear algebra: we think basis-free, but when the chips are down we close the office door and compute with matrices lie fury''

— Irving Kaplansky

speaking about Paul Halmos. These are my mathematical great-great grandfathers