Show that if all sub vectorspace are A-invariant, A must be scaled identity matrix

I'm trying to show the following:

Let $A$ be an $n \times n$ matrix over a field $F$ such that every sub vectorspace of $F^n$ is invariant with respect to $A$, then $A$ must be of the form $\lambda I_n$ with $\lambda \in F$.

Intuitively, for any $v \in F^n$, since $\operatorname{span}(v)$ is invariant with respect to $A$, it must be an eigenvector of $A$. And since this is true for all such $v$, then $A$ can only have the above form. But how do I show that last step formally?