Here we describe the Cayley-Hamilton Theorem, which states that every square matrix satisfies its own characteristic equation. This is very useful to prove results related to controllability and observability. These lectures follow Chapter 8 from: "Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control" by Brunton and Kutz Amazon: https://www.amazon.com/Data-Driven-Science-Engineering-Learning-Dynamical/dp/1108422098 Chapters available at: http://databookuw.com/databook.pdf Book Website: http://databookuw.com/ Brunton Website: eigensteve.com