Spectral Stability Under Real Random Absolutely Continuous Perturbations

Explore a probability and statistics seminar that delves into the fascinating world of random matrix phenomena and their implications for numerical linear algebra algorithms. Learn about the stability of eigenvalues and eigenvectors when adding small random variables to deterministic matrices. Discover key concepts such as tail bounds for eigenvector condition numbers and minimum eigenvalue gaps in perturbed matrices. Gain insights into Hermitization techniques, pseudospectrum analysis, and complex shifts of real matrices. Examine the challenges and surprises encountered in real-world applications of these concepts. Engage with cutting-edge research presented by Jorge Vargas, including joint work with collaborators, and consider open problems in this field of study.

Intro
The conspiracy: Machine error and non-normality
About this talk
Preliminaries: Eigenvalue condition numbers
Preliminaries: Minimum eigenvalue gap
Main theorems
Hermitization (proof technique)
Tail bounds for the singular values
The real world is challenging and full of surprises
Hermitization of the minimum eigenvalue gap
Preliminaries: Pseudospectrum
Hermitization of the eigenvalue condition numbers
Pseudospectral shattering
Complex shifts of real matrices.
Hermitization of eigenvalue condition numbers
Related work
Open problems