Matrix Inequalities in Engineering and Science

工程与科学中的矩阵不等式(DMS 9704534)李启光和罗伊·马蒂亚斯矩阵不等式理论在电气工程、统计学、经济学、心理学、图论、微分方程组、运筹学和最优化、算子论、量子物理、数值分析和控制理论等领域都有广泛的应用。然而，这些领域的许多研究人员由于他们的专业知识不是矩阵理论，而且他们与矩阵理论的研究人员之间缺乏沟通，所以无法利用矩阵理论中的现有结果和证明技术来有效地发展他们的理论。主要研究人员在矩阵不等式方面有相当丰富的经验，他们从无到有、采用或借用了许多技术来研究这一主题。在这个项目中，他们计划开发并将矩阵不等式理论应用于应用中出现的不同问题。特别是，重点将是通过概念技术而不是计算技术来证明矩阵不等式，并将它们放在更普遍和更容易获得的环境中，以便非专家可以很容易地使用结果。值得一提的是，该项目与联邦战略利益的一些领域有很强的联系。例如，在高性能计算中，需要能够高效而准确地进行计算。在许多情况下，计算对机器本身引入的舍入误差非常敏感。在这种情况下，利用矩阵不等式理论可以极大地方便快速收敛方法的设计。正如第一段中提到的，矩阵不等式是控制理论中的重要工具，对于制造工厂、飞机、汽车等的设计是至关重要的。本课程还提出了矩阵不等式在其他学科中的应用，包括量子物理和半定规划，还将考虑经济学、统计学和心理学中出现的问题。

Matrix Inequalities in Engineering and Sciences (DMS 9704534) Chi-Kwong Li and Roy Mathias The theory of matrix inequalities has applications in many fields including electrical engineering, statistics, economics, psychology, graph theory, differential equations, operations research and optimization, operator theory, quantum physics, numerical analysis and control theory. However, many researchers in these areas are not able to use the existing results and proof techniques in matrix theory to develop their theory effectively because their expertise is not in matrix theory and there is a lack of communication between them and researchers on matrix theory. The principal investigators have considerable experience in matrix inequalities and have developed from scratch, adapted, or borrowed many techniques to study the subject. In this project, they plan to develop and apply the theory of matrix inequalities to different problems that arise from applications. In particular, the emphasis will be on proving matrix inequalities by conceptual rather than computational techniques, and putting them in more general and accessible settings, so that the results can be readily used by non-specialists. It is worth mentioning that this project has strong connections with a number of areas of Federal strategic interest. For example, in high-performance computing, one needs to be able to do computation efficiently and accurately. In many situations, computations are very sensitive to round-off error introduced by the machine itself. The design of fast convergent methods in these cases is greatly facilitated by the theory of matrix inequalities. As mentioned in the first paragraph, matrix inequalities are important tools in control theory, a regime that is crucial to the design of manufacturing plants, aircraft, automobiles, etc. Applications of matrix inequalities in other subjects, including quantum physics and semi-definite programming are proposed , and problems arising in economics, statistics and psychology will also be considered.