Prove: the sum of simultaneously diagonalizable transformations is diagonalizable

Question

Let $T, S$, linear transformations which are simultaneously diagonalizable. Prove that $T+S$ is diagonalizable.

I need to rely on the the definition: $T,S$ are called simultaneously diagonalizable if there is a basis of $V$ composed by eigen-vectors of both $S$ and $T$.

I'd be glad for a direction.

Thanks.

Answer 1

Hint: Suppose that $PTP^{-1}$ and $PSP^{-1}$ are diagonal. $$ P(T + S)P^{-1} = PTP^{-1}+PSP^{-1} $$

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"How to bounce:" for a vector $v$ in our common basis of eigenvectors, $$ (T + S)v = Tv + Sv $$