Originally Posted by

**MSUMathStdnt** *In the last equal sign of the last line below, why is $\displaystyle - \lambda z = \ddot{z}$?* Thanks.

Given the DE system

$\displaystyle \ddot{\mathbf{x}}=-kA\mathbf{x}$

where $\displaystyle \mathbf{x}$ is n-vector,

$\displaystyle A$ is an *n*x*n* matrix, and

$\displaystyle \mathbf{v}$ is a an eigenvector of A,

let $\displaystyle \mathbf{x}=z\mathbf{v}$, then

$\displaystyle \ddot{\mathbf{x}}=-kA\mathbf{x}=-kA(z\mathbf{v})=-k(A\mathbf{v})z=-k(\lambda\mathbf{v})z=k(-\lambda z)\mathbf{v}=k\ddot{z}\mathbf{v}$.