= Grover iteration in a nonorthogonal basis
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{title2=$G=\begin{pmatrix}1&2a\\-2a&1-4a^2\end{pmatrix}$}
For normalized marked and uniform states $|w\rangle,|s\rangle$ with real overlap $a\in(0,1)$, the two reflections $(2|s\rangle\langle s|-I)(I-2|w\rangle\langle w|)$ have the displayed matrix in that ordered nonorthogonal basis. The coordinate <Gram matrix> is $S=\left(\begin{smallmatrix}1&a\\a&1\end{smallmatrix}\right)$ and $G^\dagger SG=S$. With $a=\sin t$, its <eigenvalues> are $e^{\pm2it}$. Applied $r$ times to $|s\rangle$, it gives marked-state success $\sin^2((2r+1)t)$. In these coordinates the overlap with the marked state is the first coefficient plus $a$ times the second; it is not generally the first coefficient alone.
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