Past exam of the numerical linear algebra and approximation course of the University of Oxford / 2008 / 2 / b

Write $G=M^{-1}N$. Subtracting the exact equation $Mx=Nx+b$ from the iteration gives $e_{\ell +1}=G{e}_{\ell }$, where $e_{\ell }=x-x^{(\ell )}$. Hence $e_{\ell }=G^{\ell }{e}_0$. Alsoso the stated scalars are precisely the eigenvalues of the iteration matrix $G$.

The spectral radius criterion for matrix powers states that $G^{\ell }\to 0$ if and only if all its eigenvalues have modulus strictly less than one. Applying that theorem gives $e_{\ell }\to 0$ for every initial vector. Therefore $\boxed{x^{(\ell )}\to A^{-1}b}$.