Speaker: Ramis Movassagh, IBM Yorktown Heights
Location: Warren Weaver Hall 1302
Date: Feb. 5, 2016, 10 a.m.
Much work has been devoted to the understanding of the motion of eigenvalues in response to randomness. The folklore of random matrix analysis, especially in the case of Hermitian matrices, suggests that the eigenvalues of a perturbed matrix repel. We prove that the complex conjugate (c.c.) eigenvalues of a smoothly varying real matrix attract. We offer a dynamical perspective on the motion and interaction of the eigenvalues in the complex plane, derive their governing equations and discuss applications. C.c. pairs closest to the real axis, or those that are ill-conditioned, attract most strongly and can collide to become exactly real. We apply the results to the Hatano-Nelson model, random perturbations of a fixed matrix, real stochastic processes with zero-mean and independent intervals and discuss open problems.