Eigenvalue

First Order Equation

Given the linear system

${\frac{d\hat{x}}{dt}= \hat{A}\hat{x}}$

and eigenvalue and corresponding eigenvector ${\lambda}$ and ${\hat{v}}$, then

${\hat{x}(t) = \hat{v}e^{\lambda t}}$

is a solution.

Second Order Equation

Given an n x n matrix ${A}$ and associated eigenvalues and eigenvectors ${-\omega_1^2, -\omega_2^2,..., -\omega_n^2}$ and ${\hat{v}_1, \hat{v}_2,..., \hat{v}_n}$, the general solution to the equation

${\hat{x}'' = \hat{A}\hat{x}}$

is given by

$\hat{x}(t) = \sum_{i=0}^{n} (c_i cos (\omega_i t) + d_i sin (\omega_i t))\hat{v}_i$