Matrix Analysis for the 21st Century

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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.