ABSTRACT
A linear map is described by a matrix, but that does not say much about its geometric properties. When you look at the 2D linear map figures from Chapter 4, you see that they all map a circle, formed from the wings of the Phoenix, to some ellipse-called the action ellipse, thereby stretching and rotating the circle. This stretching
and rotating is the geometry of a linear map; it is captured by its eigenvectors and eigenvalues, the subject of this chapter. Eigenvalues and eigenvectors play an important role in the analysis
of mechanical structures. If a bridge starts to sway because of strong winds, then this may be described in terms of certain eigenvalues associated with the bridge’s mathematical model. Figures 7.1 and 7.2 show how the Tacoma Narrows Bridge swayed violently during mere 42-mile-per-hour winds on November 7, 1940. It collapsed seconds later. Today, a careful eigenvalue analysis is carried out before any bridge is built. But bridge design is complex and this problem can still occur; the Millennium Bridge in London was swaying enough to cause seasickness in some visitors when it opened in June 2000. The essentials of all eigentheory are already present in the humble
2D case, the subject of this chapter. A discussion of the higherdimensional case is given in Section 15.1.
