Stability and instability of spectrum for noisy perturbations of non-Hermitian matrices

Abstract: We discuss the spectrum of high dimensional non-Hermitian matrices under small noisy perturbations. That spectrum can be extremely unstable, as the maximal nilpotent matrix JN with JN(i,j)=1 iff j=i+1 demonstrates. Numerical analysts studied worst case perturbations, using the notion of pseudo-spectrum. Our focus is on finding the locus of most eigenvalues (limits of density of states), as well as studying stray eigenvalues ("outliers"). I will describe the background, show some fun and intriguing simulations, and present some theorems and work in progress concerning eigenvectors. No background will be assumed. The talk is based on joint work with Anirban Basak, Elliot Paquette, and Martin Vogel.

dynamical systemsprobability

Audience: researchers in the topic

( slides | video )

Horowitz seminar on probability, ergodic theory and dynamical systems