UC Berkeley

In this thesis we look at various tools to analyse eigenvalues and eigenvectors and use themto prove the following main results. 1. We show that given any matrix A, there is a small perturbation of the matrix such that post perturbation, the matrix is almost normal. In particular there exists E, with ||E|| ≤ δ||A|| such that A + E is diagonalizable and its eigenvector matrix has polynomially (in 1/δ and n) bounded condition number. 2. We prove a necessary and sufficient condition for any local periodic operator on the universal cover of a finite graph to have a point spectrum. In particular we show that for λ to be in the point spectrum, the base graph must admit an induced forest with a very specific combinatorial structure and that the induced operator on it must also have λ as an eigenvalue. To prove the first result we study the volume of the pseudospectrum with the help of some tools from stochastic calculus. Along the way we also see why it implies a conjecture by Sankar, Spielman and Teng on the optimal constant for smoothed analysis of condition numbers. For the second result we show that a condition conjectured by Aomoto to be necessary and sufficient for the existence of point spectrum of certain operators on periodic trees is indeed so. Aomoto had already shown why the condition was necessary. We give a more intuitive proof of it and along the way also show sufficiency.