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.