Eigenvalue method for linear systems

Summary

The eigenvalue method solves a homogeneous linear system with constant coefficients $x' = A x$ with the eigenvalues and eigenvectors of $A$. If $A v = \lambda v$ with $v \ne 0$, then $x(t) = e^{\lambda t} v$ is a solution. If $A$ has $n$ linearly independent eigenvectors $v_1, \dots, v_n$ with eigenvalues $\lambda_1, \dots, \lambda_n$, then each solution is $x(t) = c_1 e^{\lambda_1 t} v_1 + \dots + c_n e^{\lambda_n t} v_n$ for unique constants $c_k$. For a complex eigenvalue $\lambda$ of a real matrix with eigenvector $v$, the real and imaginary parts of $e^{\lambda t} v$ are two real solutions.