Spectral error bound for a diagonalizable fixed-point iteration ($\Vert S^{-1}{e}_n{\Vert }_2\le \rho (G{)}^n\Vert S^{-1}{e}_0{\Vert }_2$)

Suppose $x_{n+1}=G{x}_n+c$ has a fixed point $x_{\ast }$ and $G=SD{S}^{-1}$ with diagonal $D$. For $e_n=x_n-x_{\ast }$ and $f_n=S^{-1}{e}_n$, subtraction gives $f_n=D^n{f}_0$. In the Euclidean norm,Thus $\Vert e_n{\Vert }_2\le \Vert S{\Vert }_2\Vert S^{-1}{\Vert }_2\rho (G{)}^n\Vert e_0{\Vert }_2$. 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.