Math 405 — Advanced Matrix Analysis — Spring 2013

Instructor: Mark Meckes (pronounced "MECKess")
Email: mark dot meckes at case dot edu
Office: Yost 223
Phone: 368-4997
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 quickly.)

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.

Course topics
These may be adjusted depending on time and student interest. If there's something specific you hope to see, let me know.

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 arrangements.

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.

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%.