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

正规矩阵的行为（例如，真实的对称矩阵或复厄米特矩阵）由其特征值控制;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谱上的最大绝对值。这同样适用于正规线性算子，只是现在谱可能不仅仅包括本征值。 这种说法不适用于非正规矩阵和线性算子，并且在复平面中识别可以与非正规算子相关联的集合以提供谱在正规情况下提供的信息方面有相当大的兴趣。 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，找到有效的方法来计算或近似这些集合，并将它们应用于应用数学中一些有趣的问题。特征值解释了许多不同系统的渐近行为：从流体动力学稳定性到有限差分法和迭代线性系统求解器的行为，再到概率事件的马尔可夫链建模。 然而，特征值并不能解释这些系统的瞬态行为，而这种瞬态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态行为。 这将导致更好地理解这种不同的现象，如过渡到湍流，机械或生物系统的可控性，以及从洗牌到统计力学的马尔可夫链建模中的截止行为。

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.