This chapter introduces the world of eigenvalues and eigenvectors. In German, the term eigen means characteristic or special. As we see in this chapter, eigenvalues and eigenvectors reveal key characteristics of the action of a square matrix. Eigen-analysis is very important in science and engineering. For example, eigenvalues are related to frequencies, and as an application, the stability of a bridge can be predicted by an eigenvalue analysis of a system modeling it. Figure 6.1 illustrates the collapse of the Tacoma Narrows bridge in 1940 due to wind-induced vibrations.