Matrix Exponents

Consider $D_t X = AX$ $\forall X \in M_{n \times n}(S)$. If $\vec{x} = c_1 \vec{x_1} + c_2 \vec{x_2} + \cdots + c_n \vec{x_n}$ (means $\vec{x_i}=\vec{v_i}e^{\lambda_i t}$ most likely) is a solution to $D_t \vec{x} = A \vec{x}$, then there is a solution $X=\Phi(t)=\begin{bmatrix}\vec{x_1} & \vec{x_2} & \cdots & \vec{x_n} \end{bmatrix}$ where $\vec{x_i}$ are the columns of the matrix. If there is an initial condition $\vec{x} (0) = \vec{x_0}$ then $\Phi(t) \vec{c} = \vec{x_0} \Leftrightarrow \vec{c} = \Phi ^ {-1} (t) \vec{x}$