Applied Matrix Theory and Complex Approximation: Estimating Norms of Functions of Matrices

应用数学和计算数学中的许多问题，比如微分方程或差分方程的稳定性问题，或者求解线性系统的迭代方法的收敛性问题，实际上都是关于矩阵函数的范数的问题，比如矩阵指数或矩阵幂。在实对称矩阵的情况下（或者，更一般地说，一个正常矩阵），这些问题是根据矩阵的特征值来回答的，但是当矩阵是高度非正常的（意思是它的特征向量几乎是线性相关的，或者它没有一个完整的特征向量集合），特征值只告诉故事的一部分。矩阵指数幂的渐近性质或大指数下矩阵本身的渐近性质仍然由特征值决定，但暂态性质则不是由特征值决定。在这个项目中，主要研究者将使用复平面中可以与矩阵相关联的其他集合，例如值域或ε -伪谱，来提供有关此类矩阵函数行为的更多信息，例如有限幂。在另一个方向上，关于这些矩阵函数行为的新结果将导致对复杂逼近理论问题的新见解。该奖项将支持在矩阵理论和复杂分析的数学领域的工作，这些工作在科学和工程的许多应用领域都是基础的。有一系列的数学方法可以用来描述系统的长期行为（辐射水平最终会衰减到零吗，飞机上的颤振最终会消失吗），但一个更关键的问题可能是近期会发生什么。虽然计算机模拟可以回答特定情况下的这个问题，但通常缺乏决定这些系统瞬态行为的一般结果。这个项目将为分析这类问题提供更好的数学工具。该奖项将通过研究助学金支持这些领域的研究生培训。

Many questions in applied and computational mathematics, such as questions about the stability of differential or difference equations or questions about the convergence of iterative methods for solving linear systems, are really questions about norms of functions of matrices, such as matrix exponentials or matrix powers. In the case of a real symmetric matrix (or, more generally, a normal matrix), these questions are answered in terms of the matrix eigenvalues, but when the matrix is highly nonnormal (meaning that its eigenvectors are nearly linearly dependent or it does not have a complete set of eigenvectors), eigenvalues tell only part of the story. The asymptotic behavior of powers of matrix exponentials or the matrix itself for large exponents is still determined by the eigenvalues, but transient behavior is not. In this project, the principal investigator will use other sets in the complex plane that can be associated with a matrix, such as the field of values or the epsilon-pseudospectrum, to give more information about the behavior of such matrix functions e.g. for finite powers. In the other direction, new results about the behavior of such matrix functions will lead to new insights into problems in complex approximation theory.This award will support work in the mathematical areas of matrix theory and complex analysis that is fundamental in many application areas in science and engineering. A range of mathematical methods is available for describing the long-term behavior of systems (will a radiation level eventually decay to zero, will flutter in an aircraft eventually die out), but a more crucial question may be what happens in the near-term. While a computer simulation may answer this question for a specific scenario, general results about what determines the transient behavior of such systems are often lacking. This project will provide better mathematical tools for analyzing such questions for a wide class of models. The award will support graduate student training in these areas through research assistantships.