Most of the physical systems studied involve the eigenvalues and eigenvec- tors of specific matrices. Whenever possible, I have selected systems which are described by 2 x 2 or 3 x 3 matrices. Such systems can be solved completely and are used to demonstrate the different methods of solution. In addition, these matrices will often yield the same eigenvectors for different physical systems, to provide a sense of the common mathematical basis of all the problems. For example, an eigenvector with components (1, -1) might describe the motions of two atoms in a diatomic molecule or the orientations of two atomic orbitals in a molecular orbital. The matrices in both cases couple the system components in a parallel manner.