Consider a linear transformation represented by a matrix , and some vector $v$ . If $A v = λ v$ , we say that $v$ is an eigenvector of $A$ with corresponding eigenvalue $λ$ . Intuitively, this means that $A$ doesn't rotate or change the direction of $v$ ; it can only stretch it ( $| λ | > 1$ ) or squash it ( $| λ | < 1$ ) and maybe flip it ( $λ < 0$ ). 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.