##### Primary Goal

Solve the matrix equation

Solve the matrix equation

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.

In a population of rabbits,

- half of the newborn rabbits survive their first year;
- of those, half survive their second year;
- the maximum life span is three years;
- 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?

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.