# 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.

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