## 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
$\left( \begin{array}{cc} e^{5t} & e^t \\ e^{5t} & e^{-t} \end{array} \right)$
.