= Solution
The <iteration matrix> has <eigenvalues> \b[$0,-1/2,1/2$]: expanding its <characteristic polynomial> down the first column leaves the triangular two-by-two block with diagonal entries $-1/2,1/2$. The distinct <eigenvalues> ensure <diagonalizability>. Thus its <spectral radius> is $1/2<1$.
The previous bound gives $\|f_n\|\le2^{-n}\|f_0\|\to0$, so \b[the <Gauss-Seidel iteration> converges for every initial guess] under the stated <diagonalization>. Its <limit> is the exact solution $(1,1,-1)^T$. Since $e_n=Sf_n$, a fixed <matrix> also sends this transformed error to zero; its <operator norm> can account for any amplification in the original coordinates.
Back to article page
