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The Fundamental Matrix

If  
\[M\]
  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  
\[ \left( \begin{array}{cc} e^{5t} & e^t \\ e^{5t} & e^{-t} \end{array} \right) \]
.

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