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

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 J_N with J_N(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. No background will be assumed. The talk is based on joint work with Anirban Basak and Elliot Paquette.