Math 582GB
Nonnormal Matrices and Linear Operators
Anne Greenbaum
Winter 2005, Monday/Wednesday/Friday 2:30
The spectrum of a normal matrix or linear operator (one with a complete
set of orthogonal eigenfunctions) determines much of its behavior. If $A$
is normal, then solutions to $y'=Ay$ grow or decay over time according to
whether the spectrum of $A$ extends into the right half-plane or lies
entirely in the left half plane. The 2-norms of powers of a normal
operator $A$ grow or decay with the power according to whether the
spectrum extends beyond the unit disk or is contained within the unit
disk. These statements do not hold (except asymptotically) for nonnormal
matrices and linear operators. Yet most of the operators that arise in
areas such as fluid mechanics, population dynamics, Markov chain models,
etc. are nonnormal. In this course we will explore other properties of
matrices and linear operators that can be used to gain more information
about their behavior. Concepts to be discussed include the field of
values or numerical range, the $\epsilon$-pseudospectrum, and the
polynomial numerical hull of degree $k$.
The main text for the course will be one that is soon to appear by
Trefethen and Embree entitled Spectra and Pseudospectra. The book
Topics in Matrix Analysis by Horn and Johnson will serve as a
reference, and we will also read several papers from the current
literature. There will be no exams but there will be homework problems and
a final project.
Prerequisite: Math 554 (Linear Analysis) or equivalent.