I am teaching a class in dynamical systems and I am looking for a quick proof that if a vector $v$ is in the stable subspace of a matrix $A$ and all negative eigenvalues of $A$ satisfy $\Re \lambda < - \sigma < 0$, then there is a number $K$, such that $|e^{At}v| \leq K e^{-\sigma t}|v|$.
One way to do this would be by using Jordan Canonical Form and changes of variables, another would be by proving this inequality for diagonalizable matrices and applying a density argument. I am looking for something shorter, which ideally wouldn't require Jordan Canonical Form or real analysis concepts like density.