= Linear change of coordinates
{title2=$x=Pu,\quad P\text{ invertible}$}
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 $\dot u=\alpha u-\beta v$, $\dot v=\beta u+\alpha v$.
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