Originally Posted by
Gamma I think that is supposed to be the Identity matrix?
So you need
$\displaystyle \begin{pmatrix}0 & 1 \\-1 & 0\end{pmatrix}=\begin{pmatrix}\alpha & \beta \\-\beta & \alpha\end{pmatrix}^2$
Well multiply it out and see what you get. What I got was:
$\displaystyle \alpha^2=\beta^2 \Rightarrow \alpha = +/- \beta$
and
$\displaystyle 2\alpha \beta = 1 \Rightarrow \alpha \beta = 1/2$
But then use 1 and we see we need $\displaystyle \alpha (+/- \alpha)=1/2 \Rightarrow \alpha = \frac{1}{\sqrt{2}}=\beta$