A problem on general eigenvector

Question

Suppose T$\in$$\mathcal{L}$(V),dim(V) ＝n.$λ_{i}$ and $λ_{k}$ are two different eigenvalues of T, $v_{i}$$\not=0$ is an general eigenvetor of T corresponding to $λ_{i}$,prove whether $(T-λ_{i}I)^{n}$$(T-λ_{k}I)^{n}$$v_{i}$＝0?.

Answer 1

$(T-λ_{i}I)^{n}(T-λ_{k}I)^{n}v_i=(T-λ_{k}I)^{n}(T-λ_{k}I)^{n}v_i=0,$ since $(T-λ_{i}I)^{n}$ and $(T-λ_{k}I)^{n}$ commute.