= Functions of a two-dimensional antisymmetric matrix
{title2=$A^2=-s^2I$}
Every real $2\times2$ <antisymmetric matrix> has the form $A=sJ$, where $J=\begin{pmatrix}0&1\\-1&0\end{pmatrix}$. Thus $A^2=-s^2I$, $\operatorname{tr}A=0$, and $\det A=s^2$. In particular,
$$
A^{2n}=(-s^2)^nI,\qquad A^{2n+1}=(-s^2)^nA.
$$
A convergent scalar <power series> applied to $A$ can therefore be reduced into an <even> <scalar multiple> of $I$ plus an <odd> <scalar multiple> of $A$. For real coefficients and $s\ne0$, this is $f(A)=\operatorname{Re}f(is)I+[\operatorname{Im}f(is)/s]A$, provided the series converges at $\pm is$. <Absolute convergence> justifies separating <even> and <odd> terms. This converts many small <matrix> series problems into ordinary scalar series, without computing <eigenvectors>.
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