Combinatorial matrix theory and spectral analysis

我的研究主要探讨数学的两个领域，即代数和图论的相互作用。特别关注的是被称为组合矩阵理论。我的研究的一个重要部分涉及探索图和有向图结构（例如网络风格图）如何洞察代数问题，特别是涉及矩阵特征值的见解。特征值是理解用矩阵建模的系统的长期行为的关键。 符号模式矩阵的研究起源于诺贝尔经济学奖获得者P. Samuelson的工作，但在其他建模环境中也有越来越多的兴趣。最近，模式分析已被用于检测生物和生态系统中周期性的可能性，并应用于机器学习。我不打算直接专注于一个应用程序，而是我正在进行的研究计划的重点是开发理论，可能证明在建模和算法应用程序的未来发展有用。一个长期的目标是基于上述特征值特征来表征模式的类别，但也要继续开发模式的必要条件，以允许所需的属性，特别是关注有向图条件。一个特别困难的分类是确定模式何时是潜在稳定的（只允许具有负真实的部分的本征值）或频谱任意的（对本征值没有限制）。其他类别的模式是感兴趣的，因为它们允许在动态系统中的分叉。发展我们对这些类的理解将提供对矩阵结构如何影响矩阵特征值的更深入理解。例如，进一步的发展可以提供工具来了解系统的平衡点是稳定的还是不稳定的，因此容易受到小扰动的影响。组合矩阵结构还可以让我们深入了解数学最基本的任务之一，即找到方程的解。19世纪初发现的一个特别奇怪的事实是，相对简单的五次或更高次多项式方程不允许代数解。因此，数值方法已经发展到寻找多项式方程的根。一种方法是求Frobenius伴随矩阵的特征值，例如在MATLAB软件的根命令中使用。我最近在图论方面的研究为各种新的伴随矩阵形式提供了一些见解。我计划继续探索这些结构如何为多项式的根提供新的界限，并提供可能使算法在确定多项式的根时更有效的结构，因为相应矩阵上的条件数得到了改进。

My research primarily explores the interaction of two areas of mathematics, namely algebra and graph theory. The particular focus is known as combinatorial matrix theory. A significant part of my research involves exploring how graph and digraph structures (e.g. network style diagrams) can give insight into algebraic problems, especially insights involving eigenvalues of matrices. Eigenvalues are key to understanding the long term behaviour of systems modeled with the matrix.  The study of sign pattern matrices has some of its origins in the work of Nobel Laureate P. Samuelson in economics, but has growing interest in other modeling contexts. Recently pattern analysis has been used to detect the possibility of periodicity in biological and ecological systems, and also applied in the context of machine learning. I do not plan to focus directly on an application, but instead the focus of my ongoing research program is on developing the theory that may prove useful in future development of modeling and algorithmic applications. A long term goal is characterizing classes of patterns based on eigenvalue characteristics as above, but also to continue to develop necessary conditions for a pattern to allow the needed properties, focusing especially on digraph conditions. A particularly difficult classification is the determination of when a pattern is potentially stable (allowing only eigenvalues with negative real part)  or spectrally arbitrary (putting no restrictions on the eigenvalues). Other classes of patterns are of interest as they allow for bifurcations in dynamical systems. Developing our understanding of these classes will provide a deeper understanding of how matrix structure affects eigenvalues of a matrix. As an example, further development can provide tools to understand if an equilibrium point of a system is stable or if it is unstable and hence susceptible to small perturbations. Combinatorial matrix structures can also give insight into one of the most fundamental tasks of mathematics, finding the solutions of equations. A particularly curious fact, discovered in the early 19th century, is that relatively simple polynomial equations, of degree five or higher, do not allow for algebraic solutions. Consequently, numerical methods have been developed to find roots of polynomial equations. One method finds the eigenvalues of a Frobenius companion matrix, used for example in the roots command in MATLAB software. My recent research in graph theory has provided some insight into various new classes of companion matrix forms. I plan to continue to explore how these structures can provide new bounds for roots of polynomials and provide structures that may make algorithms more efficient at determining roots of polynomials because of improved condition numbers on the corresponding matrices.