Linear mapping/Eigenvalue/Definition

Eigenvalue

Let ${}K$ be a field, ${}V$ a ${}K$-vector space and

$\varphi \colon V\longrightarrow V$

a linear mapping. Then an element ${}\lambda \in K$ is called an eigenvalue for ${}\varphi$, if there exists a vector ${}v\in V$, ${}v\neq 0$ such that

${}\varphi (v)=\lambda v\,.$