Math 405 - Matrix Analysis - Spring 2010

Instructor: Mark Meckes (pronounced "MECKess")

Email: mark dot meckes at case dot edu

Office: Yost 211

Phone: 368-4997

Class time

Currently slated for MW 2:00-3:15. If you're interested in taking the class and think that time won't work, let me know before registration starts and we may be able to change it.

Web site (this page)

http://www.case.edu/artsci/math/mwmeckes/math405/

Content and audience

This is a second course in linear algebra, geared toward students interested in functional analysis, numerical analysis, or probability, 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 and inequalities), as opposed to algebraic aspects (like multilinear and exterior algebra). 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.

Text

Matrix Analysis, by Roger Horn and Charles Johnson.

Supplemental references (I may draw some material from these, but I won't expect you to have copies of them):

Topics in Matrix Analysis by Roger Horn and Charles Johnson (the second volume of the textbook above).
Matrix Analysis by Rajendra Bhatia (this is similar in its aims to the textbook, but is more functional-analytic in spirit and requires the reader to have a more substantial background).
Matrix Computations by Gene Howard Golub and Charles F. Van Loan (a good reference for algorithmic aspects of the subject).

Course topics

• Review: vector spaces, matrices, determinants, rank, invertibility, standard inner product, similarity, eigenvalues, eigenvectors.
• Unitary equivalence, unitary triangularization, spectral theorems for Hermitian and normal matrices, QR factorization.
• Variational formulas for eigenvalues of Hermitian matrices.
• Vector and matrix norms, trace duality.
• Positive definite matrices, singular value decomposition and applications, unitarily invariant norms, Schur product theorem, positive semidefinite ordering and inequalities.
• Location and perturbation of eigenvalues.
• (If time permits) Stochastic and doubly stochastic matrices, Birkhoff's theorem and applications.

Grading

There will be weekly written homework, and mostly likely one midterm plus a final exam.