systems of first order ODEs

$${\vec{x}}^{\prime }=A\vec{x}$$

briefly we need to find all Eigenvectors:

1. find all Eigenvalues of the coefficient matrix $A$ by solving $\det (A-\lambda I)$
2. for each ${\lambda }_i$ find $(A-{\lambda }_{i}I){\vec{\xi }}_i=\vec{0}$ where $I$ is identify matrix and ${\vec{\xi }}_i$ is correspond Eigenvector
3. if the ${\lambda }_i$ is repeated eigenvalue (equivalent to repeated root), need to use $(A-{\lambda }_{i}I){\vec{\xi }}_{i+1}={\vec{\xi }}_i$ to find the generalized eigenvector, this process also called Jordan Chain
4. lastly construct the homogeneous solution $$\vec{x}(t)=\sum C_i{\xi }_i{e}^{{\lambda }_{i}t}$$