Theorems from Linear ODE systems

For a constant matrix $A$ and a non-zero vector $\mathbf u$, the function

is a solution of $\mathbf{x}'=A\mathbf{x}$ if and only if $\lambda$ is an eigenvalue of $A$ and $\mathbf{u}$ is a corresponding eigenvector. The zero solution is handled separately by taking $\mathbf u=\mathbf0$.