Let K {\displaystyle {}K} be a field, and let φ : V → V {\displaystyle {}\varphi \colon V\rightarrow V} be an endomorphism on a K {\displaystyle {}K} -vector space V {\displaystyle {}V} . Let λ ∈ K {\displaystyle {}\lambda \in K} be an eigenvalue of φ {\displaystyle {}\varphi } and P ∈ K [ X ] {\displaystyle {}P\in K[X]} a polynomial. Show that P ( λ ) {\displaystyle {}P(\lambda )} is an eigenvalue of P ( φ ) {\displaystyle {}P(\varphi )} .
![{\displaystyle {}P\in K[X]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4692e3ffcbd66e900f8975113ff3f32f1cabc21a)
