Graduate Colloquium: Simple Eigenvalues and Non-vanishing Eigenvectors of Random Matrices

Ezra Guerrero Alvarez, USC

Title: Simple Eigenvalues and Non-vanishing Eigenvectors of Random Matrices

Abstract: Given a graph, we can define a random matrix by adding a random diagonal matrix to its discrete Laplacian. We investigate when the random matrix has simple eigenvalues and its eigenvectors have no zero entries. For continuous potentials, we prove these properties hold for all but finitely many values of the coupling constant with probability 1. However, for Bernoulli potentials, the conditions fail with positive probability, for which we establish a lower bound. When the graph is a cycle of prime length, we compute the exact probability that both properties hold. This is joint work with Oluyinka Lindblad, building on ideas of Alon. No prior knowledge of spectral graph theory is assumed.