Let $V_{1}$ and $V_{2}$ be two subspaces of $\mathbb{C}^{4}$ such that $\mathbb{C}^{4} = V_{1}\oplus V_{2}$. Let $T$ be a $4\times 4$ complex matrix, viewed here as a linear operator on $\mathbb{C}^{4}$.
Question: Suppose $x$ is an eigenvector of $T$ associated to an eigenvalue $\alpha \neq 0$. Then, $Tx -\alpha x = 0$. On the other hand, if $x = x_{1}+x_{2}$, where $x_{i}\in V_{i}$, $i=1,2$, then: $$Tx -\alpha x = Tx_{1}+Tx_{2} - \alpha x_{1} - \alpha x_{2} = 0$$ Does it follow that $Tx_{1} = \alpha x_{1}$ and $Tx_{2} = \alpha x_{2}$? If not, what condition should we have if we want this condition to hold?