7. Eigenvalues and eigenvectors

When I have clarified and exhausted a subject, then I turn away from it, in order to go into darkness again.

– Carl Friedrich Gauss (1777-1855)

This section represents a culmination of nearly every topic that we have covered so far. Once we define what we mean by eigenvalues and eigenvectors of a linear transformation, we will describe the main methods used to find them.

The prefix “eigen-” comes from the German word for self. An eigenvector of a linear transformation is a non-zero vector that gets mapped to a scalar multiple of itself. In other words, an eigenvector of a linear transformation $T$ is a vector $v\ne 0$ such that $T(v)=\lambda v$ for some $\lambda \in ℝ$. Then $\lambda$ is called an eigenvalue.

The question of finding eigenvectors arises in many areas of science and statistics; for example it is commonly used in statistical analysis to convert correlated variables into uncorrelated ones. Possibly the most famous application is Google’s PageRank eigenvector, which you access everytime you use the popular search engine.