Given the following problem:
Solve the coupled equation $$\dot u = 4v,\space\space \dot v =4u$$
using the change of variables $x=u+v , \space y= u- v\space \space \space $(motivated by the eigenvectors of the coefficient marix).
I've solved it but wanted to ask: How do the eigenvectors of the coefficient matrix motivate this change of variables? Is there a general "rule" for going from the eigenvectors of the coefficient matrix to a change of variables?
The coefficient matrix is: $$\begin{bmatrix} 0 & 4 \\ 4 & 0 \end{bmatrix}$$.
Eigenvalues are 4 and -4 with eigenvectors $\{[1,0]^T, [0,1]^T\}$ and $\{[1,-1]^T\}$, respectively.
If you have a linear system $\dot w=Aw$, and a left eigenvector $\alpha$ of $A$, $αA=\lambda α$, then this applied to the ODE system gives $$ \frac{d}{dt}(αw)=αAw=λ(αw). $$ This now is one scalar differential equation in $x=(αw)$. With a basis of left eigenvectors, one can transform the system this way into a system of independent scalar DE.
Here the matrix is symmetric, the left eigenvectors are the transposes of the right eigenvectors, and $\{(1,1),(1,-1)\}$ is a basis. Thus $x=u+v$ and $y=u-v$ will lead to independent scalar DE. Their solution, transformed back to the original canonical basis, will then be solutions of the given system.