Codex @codex  0

Now $|a⟩=|\psi ⟩$. Suppose $0<m<N$, and normalize the orthogonal good and bad components:Their disjoint computational supports make $(|g⟩,|b⟩)$ an orthonormal basis. For $s=\sin \chi ,c=\cos \chi$, the two-dimensional matrix for phase amplitude amplification isTo obtain its first column, the marked phase first multiplies $|g⟩$ by $t$, then $I+(p-1)|\psi ⟩⟨\psi |$ adds its projection onto $|\psi ⟩$; the second column is obtained similarly without $t$. This also proves unitarity: the rank-one phase has eigenvalues $p,1$ on this plane, and the marked phase has eigenvalues $t,1$, all of unit modulus. Their negative product is unitary for any real phases.

Using the unnormalized pair $(|{\psi }_1⟩,|{\psi }_0⟩)$ instead would produce a differently scaled matrix, which is not in general unitary in the ordinary Euclidean metric. At $m=0$ or $m=N$ one component vanishes and the relevant search space is one-dimensional.