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Chapter5Eigenvalues and Eigenvectors

Primary Goal

Solve the matrix equation Ax = λ x .

This chapter constitutes the core of any first course on linear algebra: eigenvalues and eigenvectors play a crucial role in most real-world applications of the subject.

Example

In a population of rabbits,

  1. half of the newborn rabbits survive their first year;
  2. of those, half survive their second year;
  3. the maximum life span is three years;
  4. rabbits produce 0, 6, 8 baby rabbits in their first, second, and third years, respectively.

What is the asymptotic behavior of this system? What will the rabbit population look like in 100 years?

Figure3Left: the population of rabbits in a given year. Right: the proportions of rabbits in that year. Choose any values you like for the starting population, and click “Advance 1 year” several times. What do you notice about the long-term behavior of the ratios? This phenomenon turns out to be due to eigenvectors.

In Section 5.1, we will define eigenvalues and eigenvectors, and show how to compute the latter; in Section 5.2 we will learn to compute the former. In Section 5.3 we introduce the notion of similar matrices, and demonstrate that similar matrices do indeed behave similarly. In Section 5.4 we study matrices that are similar to diagonal matrices and in Section 5.5 we study matrices that are similar to rotation-scaling matrices, thus gaining a solid geometric understanding of large classes of matrices. Finally, we spend Section 5.6 presenting a common kind of application of eigenvalues and eigenvectors to real-world problems, including searching the Internet using Google’s PageRank algorithm.