One parameters group generated by a real matrix

Question

For a matrix $A\in M_n (\mathbb R),$ we consider the exponential $e^{tA}, t\in \mathbb R$. For $x\in \mathbb R^n\setminus\{0\},$ let $f : t\longmapsto e^{tA}x.$ My question concerns the surjectivity of the mapping $f$ from $\mathbb R$ to $\mathbb R^n$: is there a class of matrices $A$ for which $f$ is surjective for all $x\in \mathbb R^n\setminus\{0\}?$

Answer 1

Note that $A $ will always have a nonzero eigenvector, say $Ax=\lambda x $. In that case, $f (t)=e^{\lambda t}\,x ,$ so the range of $f $ is contained in a one-dimensional subspace.