= Centralizer of a planar quarter-turn
{title2=$AJ=JA\iff A=uI+vJ$}
For $J=\begin{pmatrix}0&-1\\1&0\end{pmatrix}$, the real <matrix> solutions of $AJ=JA$ are precisely
$$
A=uI+vJ=\begin{pmatrix}u&-v\\v&u\end{pmatrix},\qquad u,v\in\mathbb R.
$$
A nonzero such matrix equals $\sqrt{u^2+v^2}\,R_\theta$, where $R_\theta$ is a planar <rotation matrix> with $\cos\theta=u/\sqrt{u^2+v^2}$ and $\sin\theta=v/\sqrt{u^2+v^2}$. It acts like multiplication by the <complex number> $u+iv$. The zero matrix is the zero scalar multiple of any rotation.
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