= Solution
For the <spectral error bound for a diagonalizable fixed-point iteration>, the exact solution $x$ is a <fixed point>: $x=Gx+c$. Subtracting it from the iteration gives the error recurrence $e_n=Ge_{n-1}$, with $e_n=x_n-x$. Multiplying by $S^{-1}$ and using $G=SDS^{-1}$ gives
$$
f_n=S^{-1}GSf_{n-1}=Df_{n-1}.
$$
Iterating proves \b[$f_n=D^nf_0$], including $n=0$ because $D^0=I$. This argument does not assume $S$ is <orthogonal>; it simply expresses the error in an <eigenbasis>.
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