# Eigenvalues and eigenvectors

Consider a linear transformation represented by a matrix $$A$$, and some vector $$v$$. If $$Av = \lambda v$$, we say that $$v$$ is an eigenvector of $$A$$ with corresponding eigenvalue $$\lambda$$. Intuitively, this means that $$A$$ doesn’t rotate or change the direction of $$v$$; it can only stretch it ($|\lambda| > 1$) or squash it ($|\lambda| < 1$) and maybe flip it ($\lambda < 0$). While this notion may initially seem obscure, it turns out to have many useful applications, and many fundamental properties of a linear transformation can be characterized by its eigenvalues and eigenvectors.