Beyond Eigenvalues - Describing the Behavior of Nonnormal Matrices and Linear Operators

University Of Washington, Seattle WA

Investigators

Abstract

The behavior of a normal matrix (e.g., a real symmetric matrix or a complex Hermitian matrix) is governed by its eigenvalues; that is, the 2-norm of any analytic function of a normal matrix is just the maximum absolute value of that function on the spectrum of the matrix. The same holds for normal linear operators, except that now the spectrum may include more than just the eigenvalues. This statement does not hold for nonnormal matrices and linear operators, and there is considerable interest in identifying sets in the complex plane that can be associated with nonnormal operators to provide the sort of information that the spectrum provides in the normal case. The goal of this project is to identify such sets and determine their geometrical properties, to find efficient ways to compute or approximate these sets, and to apply them to some interesting problems in applied mathematics. Eigenvalues explain the asymptotic behavior of many different systems: from hydrodynamic stability to the behavior of finite difference chemes and iterative linear system solvers to the Markov chain modeling of probabilistic events. Eigenvalues do not explain the transient behavior of these systems, however, and it is this transient behavior that is often most important. In this work we attempt to provide the tools necessary to understand and predict this transient behavior. This will lead to better understanding of such diverse phenomena as transition to turbulence, controllability of mechanical or biological systems, and cutoff behavior in Markov chains modeling everything from card shuffling to statistical mechanics.