 Solution

For $N>1$, use the orthonormal basis $(|a⟩,|\omega ⟩)$ from the preceding part. Both reflection operators preserve $\mathcal{V}$, because their reflected vectors belong to it. With $s=\sin \theta$ and $c=\cos \theta$, their restricted matrices areMultiplying in the specified order, including the overall minus sign, givesEquivalently,In particular, $G$ sends $\sin t|a⟩+\cos t|\omega ⟩$ to $\sin (t+2\theta )|a⟩+\cos (t+2\theta )|\omega ⟩$. This proves the Grover rotation angle description with the signs appropriate to these reflection operators.