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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 5.4 we will use eigenvalues and eigenvectors to determine when a matrix is “similar” to a diagonal matrix, and we will see that the algebra and geometry of such a matrix is much simpler to understand. 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.