Let \(V\) be a vector space over a field \(F\) and let
\[\varphi \colon V\longrightarrow V\,\]
be a linear map. A vector \(v\in V\), \(v\neq 0\), is called an eigenvector of \(\varphi \) associated with the eigenvalue \(\lambda \in F\), if
\[\varphi (v)=\lambda v.\]