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.