This week we build on last week's introduction of the fundamental problem of eigenvalues and eigenvectors. Taking last week's lab as a starting point, we see that many (not all) square matrices are similar to diagonal matrices. That is, given a linear transformation (or a matrix representation), we may find a basis for the domain consisting of "favored directions" in which the effect of the transformation is just stretching or contracting -- multiplying by a scalar. When this is possible, the special basis consists of eigenvectors, and the scale factors in the corresponding directions are the eigenvalues. Furthermore, the similarity transformation that diagonalizes the matrix is determined by a matrix whose columns are the eigenvector basis. We also look briefly at the significance of having conjugate pairs of complex numbers as eigenvalues -- in this case, the geometry of the linear transformation involves a combination of rotation and scaling.
Labs this week and next week are about a significant application of eigenvalues and eigenvectors: management of an age-differentiated biological population, such as a forest, an endangered speices, or a herd of farm animals. This week we study the growth pattern of such a population, and next week we look at what happens when portions of the population are harvested periodically. This application builds on ideas first encountered in the context of Markov chains -- but here the transition matrices are Leslie growth matrices. The Markov and Leslie examples lead naturally to our study of discrete linear dynamical systems next week.
To see the syllabus for Week 11 in a separate window, click here.
Last modified: March 16, 1999