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

正规矩阵(例如，实对称矩阵或复埃尔米特矩阵)的行为由其特征值决定；也就是说，正规矩阵的任何解析函数的2-范数恰好是该函数在矩阵的谱上的最大绝对值。这同样适用于正常的线性算子，除了现在频谱可能不仅仅包括本征值之外。这一说法不适用于非正规矩阵和线性算子，人们对识别复平面中可以与非正规算子相关联的集合以提供频谱在正常情况下提供的信息有相当大的兴趣。这个项目的目标是识别这样的集合并确定它们的几何性质，找到计算或逼近这些集合的有效方法，并将它们应用于应用数学中的一些有趣的问题。本征值解释了许多不同系统的渐近行为：从流体动力稳定性到有限差分法和迭代线性系统解算器的行为，再到概率事件的马尔可夫链建模。然而，本征值并不能解释这些系统的瞬变行为，而正是这种瞬变行为往往是最重要的。在这项工作中，我们试图提供必要的工具来理解和预测这种瞬时行为。这将导致更好地理解各种现象，如向湍流的转变，机械或生物系统的可控性，以及马尔可夫链模拟从洗牌到统计力学的一切事物的切断行为。

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.