  In this work, we discuss a new method for calculation of extremal
eigenvectors and eigenvalues in systems or regions of parameter space where
direct calculation is problematic. This technique relies on the analytic
continuation of the power series expansion for the eigenvector around a point
in the complex plane. We start this document by introducing the background
material relevant to understand the basics of quantum mechanics and quantum
field theories on the lattice, how we perform our numerical simulations, and
how this relates to the nuclear physics we probe. We then move to the
mathematical formalism of the eigenvector continuation, which is rooted in
analytic function theory and linear algebra. We then discuss how these
techniques are implemented numerically, with a discussion about the
computational costs. Finally, we discuss applications of this method to full,
quantum many-body systems. These include neutron matter, the Bose-Hubbard
model, the Lipkin model, and the Coulomb interaction in light nuclei with LO
chiral forces. These systems cover two categories of interest to the field:
systems with a substantial sign problem, or systems that exhibit quantum phase
transitions.
