Mathematical Sciences: Matrix Differential Equations and Their Applications

本文主要研究矩阵微分方程的动力学及其应用。这些方程是在一个相当一般的框架中，概括了一些重要数值算法中的特殊实例。每个矩阵微分方程的动力学将通过计算机实验、高分辨率图形和符号操作来探索，并结合分析。目的是为了更好地理解矩阵微分方程的动力学，并研究与特殊结构矩阵的特征值和奇异值密切相关的问题。这项研究有望找到重要的应用范围，从开发新的计算算法到微分方程和线性代数中新问题的理论解决方案。

This research is concerned with the dynamics of matrix differential equations and their applications. The equations are cast in a fairly general framework, generalizing special instances found in some important numerical algorithms. The dynamics of each matrix differential equation will be explored via computer experiments, high resolution graphics and symbolic manipulations, in conjunction with analysis. The objective is to better understand the dynamics of matrix differential equations, and to study the closely related questions regarding the eigenvalues and singular values of specially structured matrices. This study is expected to find important applications ranging from development of novel computational algorithms to theoretic solutions of new problems in differential equations and linear algebra.