I've seen a bunch of variations on the wonderful properties of this specific matrix. My textbook gives one algebraic form in particular that I'm having a bit of trouble verifying:
Any help here? I feel like factoring out $\frac{1}{u_1^2+u_2^2}$ might be the way to start. Thanks!
Since $\left[\begin{matrix}u_1\\u_2\end{matrix}\right]$ is a unit vector, we know $u_1^2+u_2^2=1$. We have a couple things to prove: first that $2u_1^2-1=-(2u_2^2-1)$ (that is the claim that we have $a$ as the top left entry $-a$ as the bottom right entry), and second that $1=a^2+b^2=(2u_1^2-1)^2+(2u_1u_2)^2$.
Part $1$: $$u_1^2+u_2^2=1\implies u_1^2-1=-u_2^2\implies2u_1^2-2=-2u_2^2\implies 2u_1^2-1=-(2u_2^2-1)$$
as desired.
Part $2$: $$\begin{align}(2u_1^2-1)^2+(2u_1u_2)^2&=-(2u_1^2-1)(2u_2^2-1)+4u_1^2u_2^2 \\&=-(4u_1^2u_2^2-2(u_1^2+u_2^2)+1)+4u_1^2u_2^2 \\&=-(4u_1^2u_2^2-2+1)+4u_1^2u_2^2 \\&=-4u_1^2u_2^2+1+4u_1^2u_2^2 \\&=1\end{align}$$ as desired.
Pretty much any reasonable algebra manipulation should work out, just some ways are faster than others.