Let \(F\) be a field, and let \(V\) be a vector space over \(F\) and let \(\varphi \colon V\longrightarrow V\,\) be a linear function. An element \(\lambda \in F\) is called an eigenvalue of \(\varphi \), if there exist a vector \(v\in V\), \(v\neq 0\) with \[\varphi (v)=\lambda v.\]
Definitions: 1