ABSTRACT
Chapter 7 “Eigen Things” introduced the basics of eigenvalues and eigenvectors in terms of 2×2 matrices. But also for any n×nmatrixA,
are the eigenvalues. In this chapter we go a little further and examine 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, given a fun, slang name-the Google eigenvector. We explore “Eigen Things” of function spaces that are even more
general than those in the gallery in Section 14.5. “Eigen Things” characterize a map by revealing its action and ge-
ometry. This is key to understanding the behavior of any system. A great example of this interplay is provided by the collapse of the Tacoma Narrows Bridge in Figures 7.1 and 7.2. But “Eigen Things” are important in many other areas: characterizing harmonics of musical instruments, moderating movement of fuel in a ship, and analysis of large data sets, such as the Google matrix in Figure 15.1.
