ABSTRACT

A linear map is described by a matrix, but that does not say much about its geometric properties. The 2D linear map figures illustrated thus far 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. The essentials of all eigen theory are already present in the humble 2D case, the subject of this chapter. The eigendecomposition for symmetric matrices is introduced and this includes diagonalization. We demonstrate the insight that this decomposition provides and demonstrate its use in writing a matrix as a sum of rank 1 matrices. A section on quadratic forms gives deeper insight into the eigen things and characterization of a matrix. We also look at this in terms of repeated application of a linear map. Sketches and figures illustrate the concepts.