Past exam of the year 1 of the computer science course of the University of Oxford / 2019 / Digital systems, linear algebra, introduction to formal proof / 6 / a / v

The iteration matrix has eigenvalues $0,-1/2,1/2$: 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 $\Vert f_n\Vert \le 2^{-n}\Vert f_0\Vert \to 0$, so the Gauss-Seidel iteration converges for every initial guess under the stated diagonalization. Its limit is the exact solution $(1,1,-1{)}^T$. Since $e_n=S{f}_n$, a fixed matrix also sends this transformed error to zero; its operator norm can account for any amplification in the original coordinates.