In this thesis I investigate the utility of the polarization and angle dependence of the photoemission matrix element in angle resolved photoemission spectroscopy (ARPES). This technique is capable of determining internal symmetries of the electronic wave functions in crystalline solids and has been historically underutilized in the ARPES community. In Chapter 1, I introduce the ARPES technique, the established theory and models behind it, and the experimental considerations in performing the technique. It is my personal belief that the fastest way to build intuition for complex physical phenomena is through simulation, and a good portion of graduate school career was spent developing simulation toolkits for the photoemission process. This work will be covered in detail in chapter 2. The later chapters cover the application of this technique to various materials systems. Chapter 3 focuses on the use of this technique to observe a topological phase transition in the Lanthanum Monopnictides. Chapter 4 contains my work on the Cuprate high temperature superconductors, Bi2Sr2Ca1Cu2O8+ x and La2--xSr xCuO2 and the use of tight binding simulations to approximate the single particle wave function in this material.