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