Take the \(z\)-transform of (6.2.6) and show that \(\mathbf{X}(z)=\mathbf{G}(z) \mathbf{Y}(z)\), where \(\mathbf{G}(z)=(z \mathbf{U}-\mathbf{M})^{-1} \mathbf{D}^{-1}\) and \(\mathbf{U}\) is the unit matrix. \(\mathbf{G}(z)\) is the matrix transfer function of a digital filter that represents the Jacobi or Gauss-Seidel methods. The filter poles are obtained by solving \(\operatorname{det}(z \mathbf{U}-\mathbf{M})=0\). The filter is stable if and only if all the poles have magnitudes less than 1.

\[
\begin{equation*}
\mathbf{x}(i+1)=\mathbf{M x}(i)+\mathbf{D}^{-1} \mathbf{y} \tag{6.2.6}
\end{equation*}
\]