Functions of a two-dimensional antisymmetric matrix ($A^2=-s^{2}I$)

Every real $2\times 2$ antisymmetric matrix has the form $A=sJ$, where $J=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$. Thus $A^2=-s^{2}I$, $\mathrm{tr}A=0$, and $\det A=s^2$. In particular,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 $s\ne 0$, this is $f(A)=\mathrm{Re}f(is)I+[\mathrm{Im}f(is)/s]A$, 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.