$T, S : V \to V$ linear transformations such that $S◦T = T◦S$. Prove that every eigenspace of $T$ is $S$-invariant.

Question

$T, S : V \to V$ linear transformations such that $S◦T = T◦S$. Prove that every eigenspace of $T$ is $S$-invariant.

I have seen this claim here - $\mbox{Ker} \;S$ is T-invariant, when $TS=ST$ but can't quite understand how to prove it.

Any help would be appreciated.

Answer 1

Suppose v is an eigenvector of T. Then there is some scalar $\lambda$ such that Tv = $\lambda$v. Scalars commute with everything, so STv = S$\lambda$v = $\lambda$Sv.

Since ST = TS, STv = TSv, so $\lambda$Sv = TSv

That can be rewritten as T(Sv) = $\lambda$(Sv)

In other words, Sv is an eigenvector of T, with the same eigenvalue as v.