For my senior capstone project as a Mathematics major at Knox, I completed a six month study on matrix Lie groups (specifically focusing on the rotation Lie groups SU(n) and SO(n)) and their usage in quantum mechanics and general relativity, culminating in a public presentation and a one-hundred page paper. None of the research in this paper is my own but the way that it's presented here, along with a few of the connections made, are my own. I wanted to make it publicly available to anyone interested so, here it is!