= Spectral error bound for a diagonalizable fixed-point iteration
{title2=$\|S^{-1}e_n\|_2\le\rho(G)^n\|S^{-1}e_0\|_2$}
Suppose $x_{n+1}=Gx_n+c$ has a <fixed point> $x_*$ and $G=SDS^{-1}$ with diagonal $D$. For $e_n=x_n-x_*$ and $f_n=S^{-1}e_n$, subtraction gives $f_n=D^nf_0$. In the <Euclidean norm>,
$$
\|f_n\|_2^2=\sum_i|\lambda_i|^{2n}|(f_0)_i|^2\le\rho(G)^{2n}\|f_0\|_2^2.
$$
Thus $\|e_n\|_2\le\|S\|_2\|S^{-1}\|_2\rho(G)^n\|e_0\|_2$. <Spectral radius> below one implies <convergence> for every start. The unit-constant bound belongs to the transformed coordinates; it does not automatically hold in the original <Euclidean norm> when $S$ is nonorthogonal.
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