Chapter 7
Eigen Things
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 ...
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