\[M\]
is a matrix of coefficients for a system of a linear system of coupled linear differential equations \[\dot{\mathbf{v}}=M \mathbf{v}\]
with eigenvalues \[\lambda_1, \: \lambda_2,..., \lambda_n\]
with corresponding eigenvectors \[\mathbf{v}_1, \: \mathbf{v}_2,..., \: \mathbf{v}_n\]
then \[X(t)=(\mathbf{v}_1 e^{\lambda_1} , \: \mathbf{v}_2 e^{\lambda_2} ,..., \: \mathbf{v}_n e^{\lambda_n})\]
Example: The system
\[\dot{x}=3x+2y\]
\[\dot{y}=2x+3y\]
has eigenvalues
\[\lambda_1=5, \lambda_2=1\]
with corresponding eigenvectors \[\begin{pmatrix}1\\1\end{pmatrix}, \: \begin{pmatrix}1\\-1\end{pmatrix}\]
.A fundamental matrix is
\[ \left( \begin{array}{cc} e^{5t} & e^t \\ e^{5t} & e^{-t} \end{array} \right) \]
.