Applied Matrix Theory and Complex Approximation: Estimating Norms of Functions of Matrices

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Many questions in applied and computational mathematics, such as questions about the stability of differential or difference equations or questions about the convergence of iterative methods for solving linear systems, are really questions about norms of functions of matrices, such as matrix exponentials or matrix powers. In the case of a real symmetric matrix (or, more generally, a normal matrix), these questions are answered in terms of the matrix eigenvalues, but when the matrix is highly nonnormal (meaning that its eigenvectors are nearly linearly dependent or it does not have a complete set of eigenvectors), eigenvalues tell only part of the story. The asymptotic behavior of powers of matrix exponentials or the matrix itself for large exponents is still determined by the eigenvalues, but transient behavior is not. In this project, the principal investigator will use other sets in the complex plane that can be associated with a matrix, such as the field of values or the epsilon-pseudospectrum, to give more information about the behavior of such matrix functions e.g. for finite powers. In the other direction, new results about the behavior of such matrix functions will lead to new insights into problems in complex approximation theory.This award will support work in the mathematical areas of matrix theory and complex analysis that is fundamental in many application areas in science and engineering. A range of mathematical methods is available for describing the long-term behavior of systems (will a radiation level eventually decay to zero, will flutter in an aircraft eventually die out), but a more crucial question may be what happens in the near-term. While a computer simulation may answer this question for a specific scenario, general results about what determines the transient behavior of such systems are often lacking. This project will provide better mathematical tools for analyzing such questions for a wide class of models. The award will support graduate student training in these areas through research assistantships.