Math 407/507
Applied Linear Algebra
Fritz Keinert
Fall 2016

Announcements [old announcements]

  • [Dec 13] I have posted the answers to the final exams, and the final scores, in BlackBoard. I will be out of town Wednesday, Dec 14, but back on Thursday, if you want to pick up your exam.
  • [Dec 11] I plan to be around for my usual 11-12 Monday office hour during finals week.
  • [Dec 9] I have posted the answers to HW 4 in BlackBoard, and finished updating all the lecture notes below. (The scanner is acting up: there is a line across every page).
  • [Dec 7] I have created a page with topics on the final exam, and a file with review problems.

Homework Assignments

The links will become active later.

HW 1, due Friday, Sept 16 [elnino.txt]
HW 2, due Friday, Oct 7
Midterm on Monday, Oct 10
HW 3, due Friday, Nov 11 [mona_small.gif]
HW 4, due Friday, Dec 2
Final Exam Tuesday, Dec 13, 9:45-11:45am


Fritz KeinertFritz Keinert
464 Carver
Office hours: MWF 11am-noon, and by appointment.
Class meetings: MWF 10:00-10:50am, Carver 290

About Math 407/507

(3-0) Cr. 3. F.
Prereq: MATH 207 or MATH 317
Advanced topics in applied linear algebra including eigenvalues, eigenvalue localization, singular value decomposition, symmetric and Hermitian matrices, nonnegative and stochastic matrices, matrix norms, canonical forms, matrix functions. Applications to mathematical and physical sciences, engineering, and other fields.

There will be some overlap between this course and Math 510 (Theoretical Linear Algebra) and 562 (Numerical Linear Algebra). Compared to Math 510, we will not do a lot of proofs and concentrate more on applications. Compared to Math 562, we will talk about the ideas underlying some of the algorithms, but not derive or study them in detail.

I will try to present interesting applications of linear algebra in various fields, but additional suggestions from the students are welcome.

The difference between Math 407 and Math 507 is that some of the homework problems (especially the more theoretical ones) will be assigned for Math 507 students only.


There is no official textbook. I will make my lecture notes available.There is one book that was published recently that looks very interesting. I may base some of my lectures on it, and there is a copy of it on reserve at the library.

book coverHelen Shapiro
Linear Algebra and Matrices:
Topics for a second course
ISBN: 978-1-4704-1852-6,
or through the AMS bookstore
(there is also an electronic version)

This book is based on topics from two courses, so it is too long to cover in one semester. I plan to use some of the content from chapters 1-8 and 15-19.

Homeworks, Exams, Grades

I will assign 4 homeworks, which together will be worth 50% of your grade. The homeworks will not necessarily all carry the same number of points. Some of the harder problems will be for the Math 507 students only.

There will also be a midterm and a final exam, each worth 25% of your score.


There is a page for this course in BlackBoard. That page is for posting solutions to homework problems, scores, and other things I want to hide from Google. Other than that, there is nothing of interest there.

Matlab and Sage

You will need a system that can calculate with matrices. You can use whatever you like, but if you want help from me, it should be Matlab.

This is like using a calculator in Calculus. You can use one from Hewlett-Packard, or Texas Instruments, or Casio, or whatever, graphing or not. All that matters is that you know how to use it, and you get some numbers out of it.

Alternatives to Matlab include Mathematica, Maple, and Sage. These are mostly programs for symbolic calculations, but they can handle numerical values as well. Sage can be found at One of the previous instructors (Prof. Hogben) worked a lot with Sage. I am not really familiar with it. Over the years, I have seen other creative approaches, such as Excel, Autocad, and maybe a few more.

Outline of the semester

The following outline is tentative. Feel free to suggest more topics that interest you.

Week and Dates Material
Week 1 (Aug 22-26) Review of Linear Algebra
Intro to Matlab
(Shapiro ch. 1)
Week 2 (Aug 29-Sep 2) Inner Products, Orthogonality
Fourier Series
Gram-Schmidt Orthonormalization
QR Decomposition
Week 3 (Sep 5-9) Least Squares Problems
Compressed Sensing
Week 4 (Sep 12-16) Eigenvalues
Vector and Matrix Norms
Condition Number
Week 5 (Sep 19-23)

Eigenvalue Localization
Nilpotent Matrices

Week 6 (Sep 26-30) Jordan Normal Form
Symbolic Math in Matlab
Computer Experiments
Companion Matrix
Week 7 (Oct 3-7) Matrix Factorizations,
in particular Schur and SVD
Week 8 (Oct 10-14) Midterm
Moore-Penrose Generalized Inverse
Week 9 (Oct 17-21) Matrix Exponentials and other Matrix Functions
Matlab Demo
Week 10 (Oct 24-28) Application to ODEs
Week 11 (Oct 31-Nov 4) Convergence of Anx
Non-negative Matrices
Stochastic Matrices
Markov Chains
Week 12 (Nov 7-11)

Applications to Graph Theory
Directed Graphs

Week 13 (Nov 14-18)

Linear Programming
Compressed Sensing

Thanksgiving Break
Week 14 (Nov 28-Dec 2) Error-correcting Codes
Week 15 (Dec 5-9) Fast Matrix Multiplication
Finals Week (Dec 12-16) Final
Tuesday, 9:45-11:15am

Course Content

This course will feature a review of known topics from basic linear algebra, as well as some new theory, followed by a variety of applications in various fields. The contents of the course are expected to change every semester, depending on the background and interests of whoever is teaching it. The following list describes my personal plans for this course.

  • Theory: Review of basic linear algebra; introduction to Matlab
  • Theory: Inner products; orthogonality, orthogonal and unitary matrices
  • Applications: Gram-Schmidt orthonormalization; QR Factorization; Fourier Series; overdetermined least squares problems.
  • Theory: Vector and matrix norms; condition numbers
  • Theory: Eigenvalues; Gershgorin theorem; spectral radius; Jordan normal form; nilpotent matrices; Schatten norms
  • Applications: Convergence of A^n x; matrix exponentials and other matrix functions; use of companion matrix to solve polynomial equations
  • Theory: Normal, Hermitian and unitary matrices; positive definite matrices
  • Applications: More matrix factorizations, in particular SVD; applications of SVD
  • Applications: Use of matrices in graph theory
  • Theory: Non-negative matrices; stochastic matrices
  • Applications: Markov chains
  • Applications: Error-correcting codes; fast matrix multiplication

Official Math Department Policies

The Math Department Class Policies page describes the official policies that all instructors have to follow. It covers rules on make-up exams, cheating, student behavior, etc.

Students With Disabilities

If you have a documented disability and require accommodations, you should obtain a Student Academic Accommodation Request (SAAR) from the Disability Resources office (Student Services Building, Room 1076, 294-6624 or TDD 294-6335, or Please contact your instructor early in the semester so that your learning needs may be appropriately met.

More details can be found in the Math Department Disability Accommodation Policy.

Last Updated: December 13, 2016