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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., Ph.D. Dissertation. Submitted to Michigan State University. Supervised by Dean J. Lee. 118 pages, 63 figures