Suppose I have the following set of differential equations

$\dot x\_1 = -x\_1 - 3x\_2 \quad \dot x\_2 = 2x\_2$

This is the phase portrait of our system
[![enter image description here](https://i.stack.imgur.com/lilh9.png)](https://i.stack.imgur.com/lilh9.png)

But let's take a change of coordinate using $P = \begin{bmatrix} 1 & 0 \\ -1 & 1 \end{bmatrix}$ where each of the columns corresponds to an eigenvector

Then let $y = P^{-1}x$
[![enter image description here](https://i.stack.imgur.com/s6TpE.png)](https://i.stack.imgur.com/s6TpE.png)

Couple things:

1. The phase portrait has been "straightened"
2. The system of equation becomes decoupled (i.e. only diagonal elements exist)

What accounts for this amazing behavior? Why is it this particular change of coordinate matrix is able to yield a new set of equations with these properties?