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.

Remarks

Example: $A = \begin{pmatrix} 1 & 1 \\ 4 & 1 \end{pmatrix}$ has the eigenvalues $3$ and $-1$, with eigenvectors $(1, 2)$ and $(1, -2)$. So $x(t) = c_1 e^{3t} (1, 2) + c_2 e^{-t} (1, -2)$. If $A$ has too few eigenvectors, generalized eigenvectors give solutions of the form $e^{\lambda t}$ times a polynomial in $t$ with vector coefficients. The eigenvalues decide the stability of the system.

Explanations

L1 human

Along an eigenvector, the matrix $A$ acts as multiplication by a number. So on this line the system is the simple equation $y' = \lambda y$, with solution $e^{\lambda t}$. Write the start vector in a basis of eigenvectors, and let each part grow or decay at its own rate.