Abstract: Eigenvalues provide a crucial tool for analyzing the asymptotic stability of linear (or linearized) dynamical systems. However, many systems that involve nonsymmetric matrices can exhibit a transient phase before the asymptotic stability is realized; indeed, during this transient phase solutions that eventually converge to zero can stagnate or even grow by orders of magnitude. Such transient behavior has significant implications for linear stability analysis (e.g., for assessing stability of a steady-state fluid flow to small perturbations), as well as for the convergence rate of numerical methods (e.g., delaying the convergence of the GMRES method for solving Ax=b beyond what one would expect from eigenvalues alone).
In this talk, we will describe the mechanism for this transient growth (non-orthogonality of eigenvectors), and show its implications for a range of settings, including solutions of differential algebraic equations and the convergence of linear solvers.
Bio: Mark Embree is a Professor of Mathematics at Virginia Tech. A graduate of Virginia Tech, he completed his doctorate in Numerical Analysis at Oxford University. From 2002-2013, he was on the faculty of Computational and Applied Mathematics at Rice University. Since 2015 he has led Virginia Tech’s undergraduate major in Computational Modeling and Data Analytics, which has graduated over 600 students.