Linear change of coordinates ($x=Pu,P\text{ invertible}$)

An invertible linear transformation replaces one system of coordinates by another. If $x=Pu$ in a linear differential equation $\dot{x}=Ax$, then $\dot{u}=P^{-1}APu$. The similar matrices $A$ and $P^{-1}AP$ have the same characteristic polynomial and eigenvalues, although their coordinate descriptions of the eigenvectors differ. Such a change can turn a planar linear system with eigenvalues $\alpha \pm i\beta$ into $\dot{u}=\alpha u-\beta v$, $\dot{v}=\beta u+\alpha v$.