# 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.