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.