Math 405 — Advanced Matrix Analysis —
Mark Meckes (pronounced "MECKess")
Email: mark dot meckes
Office: Yost 223
Office hours: MWF 1:30–2:30 p.m.
Class time and location
MWF 10:30–11:20 in Yost 101.
Web site (this page)
Content and audience
This is a second course in linear algebra, geared toward students
interested in numerical analysis, functional analysis, probability, or
statistics, or anyone else interested in more advanced topics in
matrix theory. As the course title suggests, the emphasis will be
more on analytic aspects of matrix theory (things like variational
principles, norms, and inequalities), as opposed to algebraic aspects
(like canonical forms and multilinear algebra). If there are specific
topics you'd like to see, let me know.
I will assume students are already familiar with (real and complex)
vector spaces, inner products, linear transformations, matrices and
matrix algebra, determinants, and eigenvalues and eigenvectors.
(These ideas will be reviewed at the beginning of the course, but very
Official course description
An advanced course in linear algebra and matrix theory. Topics
include variational characterizations of eigenvalues of Hermitian
matrices, matrix and vector norms, characterizations of positive
definite matrices, singular value decomposition and applications,
perturbation of eigenvalues. This course is more theoretical than
Math 431, which emphasizes computational aspects of linear algebra.
Prerequisite: Math 307, or an equivalent
undergraduate background in linear algebra.
Textbook: Matrix Analysis, second edition, by
Roger Horn and Charles Johnson.
You may have seen announcements for this course stating that
there would be no official textbook. After seeing the new second
edition, I decided that it would be an appropriate textbook, and I
will follow it loosely in the course. However, some course
topics do not appear in the textbook and some may be covered
rather differently than in the book.
These may be adjusted depending on time and student interest. If
there's something specific you hope to see, let me know.
- Review: vector spaces, matrices, determinants, rank,
invertibility, standard inner product, similarity, eigenvalues,
- Triangularization, spectral theorems, singular value decomposition,
and other matrix factorizations.
- Differential analysis of spectra.
- Variational principles and inequalities for eigenvalues.
- Vector and matrix norms
- Locations of eigenvalues
- Matrices with positive entries
- Markov chains
Homework There will be homework
problems based on the material from each lecture, normally due
in class on Friday. Late homework will not be accepted. If
unusual circumstances arise and you contact
me in a timely manner, then we can discuss alternative
Throughout this class, you need to justify your answers
even when the problem doesn't explicitly ask for
explanation; this typically means writing in
complete English sentences. When deciding how much
detail to include, here's the standard to keep in mind: your
solution to a problem should be complete and clear enough
that one of your classmates, who has paid attention in class
but hasn't thought about that specific problem yet, could
read your solution and understand exactly how it works. If
you only try to convince me that you understand the
solution, then you almost certainly won't write enough.
Students may work together on homework. However, each
student must figure out how to write up his or her own
solution to be turned in. That means for example that you
and a friend may figure out together how to prove a
statement, but the written-out proofs you turn in should not
be verbatim copies of each other.
Reading and homework problems will be posted
on this page.
Exams There will be four in-class
midterm exams, on February 8, March 4, April 5, and April 29.
The final exam will be May 2, 8:30–11:30 a.m. All exams
are closed-book and closed-notes with no calculators allowed.
Your grade for the semester will be computed as follows:
Homework 40%; each midterm exam 10%; final exam 20%.