Methods from Linear ODE systems

When $A$ has an eigenbasis $v_1,\ldots,v_n$ with eigenvalues $\lambda_1,\ldots,\lambda_n$:

1. find the eigenvalues and corresponding eigenvectors;
2. write the initial vector as $\mathbf{x}_0=c_1v_1+\cdots+c_nv_n$;
3. replace each eigenmode $v_j$ by $e^{\lambda_jt}v_j$;
4. combine the modes: $$\mathbf{x}(t) =c_1e^{\lambda_1t}v_1+\cdots+c_ne^{\lambda_nt}v_n.$$