Two-dimensional matrix for phase amplitude amplification ($Q=-[I+(e^{i\varphi }-1)|\psi ⟩⟨\psi |]\mathrm{diag}(e^{i\theta },1)$)

If the state-preparation unitary sends $|0⟩$ to $|\psi ⟩=\sin \chi |g⟩+\cos \chi |b⟩$, use the orthonormal good/bad basis. The marked-space phase is diagonal there, and conjugating a phase on $|0⟩$ gives the rank-one phase in the displayed formula. Their product is unitary. An arbitrary preparation unitary not sending zero to this state need not preserve this particular two-dimensional subspace. Matrices in unnormalized good/bad vectors must not be described as unitary in the ordinary Euclidean inner product.