For any n x n matrix A, we may ask if it has fixed directions, eigenvectors, and what are the corresponding eigenvalues. Once we review the basics, we will introduce new properties of “eigen things” and tools that will prepare us for algorithms to approximate eigenvalues and eigenvectors. Similarity and diagonalization are examined, as well as quadratic forms. The first method we introduce is the power method for finding the eigenvector that corresponds to the dominant eigenvalue. This method is paired with an application section describing how a search engine might rank webpages based on this special eigenvector. The second method we introduce for approximating eigenvalues is the QR algorithm, based on the QR decomposition. We explore eigen things of function spaces that are even more general than those in the gallery in Chapter 14. We pair this idea with an application of modelling influenza. Eigen things characterize a map by revealing its action and geometry. This is key to understanding the behavior of any system. Sketches and figures illustrate the concepts.