Homogeneous linear systems: constant coefficients

Homogeneous linear systems of ODEs take the following form:

${\vec {x}}'=Ax$

Similar to the scalar case, we look for some exponential solutions:

${\vec {x}}=e^{rt}{\vec {\xi }}$

Doing some basic differentiation and substitution from the above two equations, we get

$r{\vec {\xi }}=A{\vec {\xi }}$

Notice that \xi is an eigenvector. The entire eigenspace is solutions.