Linear Algebra with Applications, 10th Edition

6.1 Eigenvalues and Eigenvectors

Many application problems involve applying a linear transformation repeatedly to a given vector. The key to solving these problems is to choose a coordinate system or basis that is in some sense natural for the operator and for which it will be simpler to do calculations involving the operator. With respect to these new basis vectors (eigenvectors), we associate scaling factors (eigenvalues) that represent the natural frequencies of the operator. We illustrate with a simple example.

Example 1

Let us recall Application 1 from Section 1.4. In a certain town, 30 percent of the married women get divorced each year and 20 percent of the single women get married each year. There are 8000 married women and 2000 single ...