  Toeplitz matrices have entries that are constant along diagonals. They model
directed transport, are at the heart of correlation function calculations of
the two-dimensional Ising model, and have applications in quantum information
science. We derive their eigenvalues and eigenvectors when the symbol is
singular Fisher-Hartwig. We then add diagonal disorder and study the resulting
eigenpairs. We find that there is a "bulk" behavior that is well captured by
second order perturbation theory of non-Hermitian matrices. The
non-perturbative behavior is classified into two classes: Runaways type I leave
the complex-valued spectrum and become completely real because of eigenvalue
attraction. Runaways type II leave the bulk and move very rapidly in response
to perturbations. These have high condition numbers and can be predicted.
Localization of the eigenvectors are then quantified using entropies and
inverse participation ratios . Eigenvectors corresponding to Runaways type II
are most localized (i.e., super-exponential), whereas Runaways type I are less
localized than the unperturbed counterparts and have most of their probability
mass in the interior with algebraic decays. The results are corroborated by
applying free probability theory and various other supporting numerical
studies.
