Prove, that $\vec{\phi }(t)=e^{\lambda_0t} \cdot \vec{x}_0 $ is a solution to the system $X'=AX$

Question

If $\lambda_0$ is an eigenvalue of matrix $A$, corresponding to eigenvector $\vec{x}_0$, then vectorfunction $\vec{\phi}(t)=e^{\lambda_0 t} \cdot \lambda_0 \vec{x}_0$ is a solution to homogeneous system $~X'=AX~$. Do, someone know the name of the theorem and if possible where to find the proof with detailed explanations?

Answer 1

If $v$ is an eigenvector of $A$ with eigenvalue $\lambda$, then $$ X(t) = e^{\lambda t} v \implies X'(t) = e^{\lambda t} (\lambda v) = e^{\lambda t} (Av) = A (e^{\lambda t} v) = AX(t), $$ so $X$ solves the differential equation.