Matrix Analysis for the 21st Century

在飞机或工业设施等系统的设计中，必须知道设计是最佳的和安全的；通常，这种稳定性是用某个矩阵方程的“正性”来表示的。更一般地说，矩阵不等式在工程和其他应用中非常重要，因为系统的稳定性通常可以用一组复杂的矩阵不等式来表示。矩阵分析的数学学科可以用来简化或更好地理解关于这种不等式的问题。这个项目将继续发展矩阵不等式的系统操作，并研究几个复杂变量的相关数学问题，其中许多是独立的理论兴趣。该项目还将探索数学以外的潜在应用，包括工程和经济学的应用，例如寻找贸易限制模型中的不稳定均衡，这可能为贸易政策提供信息。矩阵不等式建立了一个包含若干矩阵的表达式的特征值的正性。例如，在常微分方程系统的稳定性分析中出现矩阵不等式，其中需要Lyapunov条件的矩阵解。这个项目将有助于矩阵不等式的系统代数和分析操作的数学基础。该学科最近的发展涉及到自由非交换函数的研究，这是一种用于以无维方式执行此类操作的自然函数类。对自由非交换泛函演算的定性理解对于应用是重要的；它是一种工具，用于解决矩阵计算（例如求逆和乘法）在计算上昂贵且有时不稳定的问题。许多研究应用了系统和控制工程中发展起来的技术，如实现理论和平方和，这些技术的数学理论目前正处于快速发展的阶段。本项目还涉及解析函数的边界行为，自Nevanlinna和Loewner的经典工作以来，解析函数在矩阵不等式和矩理论的处理中发挥了重要作用。该研究在自由概率论、随机矩阵理论、若干复变量和实际代数几何等方面具有潜在的应用前景。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

In the design of systems such as aircraft or industrial facilities, it is essential to know that the design is optimal and safe; oftentimes such stability is encoded in terms of the "positivity" of a certain matrix equation. More generally, matrix inequalities are of great importance in engineering and other applications, since the stability of a system can often be expressed in terms of a complex set of matrix inequalities. The mathematical discipline of matrix analysis can be used to simplify or better understand questions about such inequalities. This project will continue the development of the systematic manipulation of matrix inequalities and study related mathematical questions in several complex variables, many of which are of independent theoretical interest. This project will also explore potential applications outside mathematics, including applications to engineering and economics, such as a search for unstable equilibria in models of trade restrictions, which may inform trade policy. A matrix inequality establishes the positivity of the eigenvalues of an expression involving some matrices. For example, matrix inequalities arise in the stability analysis of systems of ordinary differential equations where a matrix solution to the Lyapunov condition is needed. This project will contribute to the mathematical foundations of systematic algebraic and analytic manipulation for matrix inequalities. Recent development of the subject has concerned the study of free noncommutative functions, the natural class of functions used to perform such manipulations in a dimension-free way. A qualitative understanding of the free noncommutative functional calculus is important for applications; it is a tool to work around the fact that matrix calculations, such as inversion and multiplication, are computationally expensive and sometimes unstable. Much of the study applies techniques developed in systems and control engineering, such as realization theory and sums of squares, whose mathematical theory is currently undergoing a rapid development. This project is also concerned with the boundary behavior of analytic functions, which has played an important role in the manipulation of matrix inequalities and moment theory since the classical work of Nevanlinna and Loewner. The research has several potential applications to free probability, random matrix theory, several complex variables, and real algebraic geometry.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.