Eigenvalues and eigenvectors/Homothety/Remark

In case of a homothety with factor ${}a$, every vector ${}v\neq 0$ is an eigenvector for the eigenvalue ${}a$. The eigenspace of the eigenvalue ${}a$ is the total space. Also, we can restrict an endomorphism to any eigenspace (as source and target), yielding the mapping

$\varphi {|}_{\operatorname {Eig} _{\lambda }{\left(\varphi \right)}}\colon \operatorname {Eig} _{\lambda }{\left(\varphi \right)}\longrightarrow \operatorname {Eig} _{\lambda }{\left(\varphi \right)}.$

This mapping is just the homothety with factor ${}\lambda$.