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.