M.Sc Student M.Sc Thesis Weintroub Amit On the Angles between Eigenvectors in the Complex Ginibre ensemble Department of Mathematics ASSOCIATE PROF. Ron Rosenthal ASSOCIATE PROF. Nicholas Crawford

Abstract

A complex Ginibre matrix is an N?N non-Hermitian random matrix over the complex numbers with independent identically distributed centered complex Gaussian entries. In contrast to the situation in random Hermitian matrices, in which the eigenvectors are orthogonal, in the non-Herimitian case, the eigenvectors are non-orthogonal and highly depend on the eigenvalues. Unlike the situation in the Hermitian case in which both the eigenvalues and the eigenvectors are known, in the non-Hermitian case, there are still many open questions regarding the correlations between the eigenvectors. In this manuscript, we study several aspects related to the angles between pairs of distinct eigenvectors of a complex Ginibre matrix in the limit as N → ∞, and thus obtain new information on the correlation structure of these eigenvectors. In particular, we calculate the limiting distribution of the angle between pairs of typical eigenvectors and give a closed formula for the limiting distribution, provide estimation on the maximal angle between pairs of distinct eigenvectors, and quantify the relation between the eigenvalues distance and the size of the angle obtained by their corresponding eigenvectors. We conclude the manuscript by discussing plans for further work on remaining open problems and by providing some data originating in related Matlab simulations.