# Jacobi and Gauss-Seidel convergence of a Matrix

Let

$$B = \begin{pmatrix} \epsilon &0 &1 \\0& \epsilon& 0 \\ 1 &0 & \epsilon\end{pmatrix}$$

For which $\epsilon \in \mathbb R\setminus\{0\}$ do the Jacobi-method and Gauss-Seidel-method converge?

First of all B is irreducible . Then i used the weak row-sum criterion

$\quad (1) \quad \sum_{i \neq j} |a_{ij}| \leq |a_{ij}| \quad(i=1,...n)$

$\quad (2) \quad \sum_{i \neq j} |a_{ij}| \lt |a_{ii}|$ for at least one row

This works for all $|\epsilon| \gt 1$. That implies the spectral radius is $\lt$ 1 and therefore both methods converge. Is this the correct answer or am I missing something?

Let $B = D - L - U$ be the usual decomposition of $B$ into a diagonal matrix $D$, a strictly lower triangular part $L$ and a strictly upper triangular part $U$.
Jacobi's method has $$x_{n+1} = G x_n + f,$$ where $$G = D^{-1}(L + U) = \begin{bmatrix} 0 & 0 & -\epsilon^{-1} \\ 0 & 0 & 0 \\ -\epsilon^{-1} & 0 & 0 \end{bmatrix}$$ It follows, that $\|G\|_\infty = \epsilon^{-1}$ and the iteration converges for all $f$ if and only if $\epsilon > 1$.
Gauss-Seidels method has \begin{align} G &= (L - D)^{-1} U \\&= \begin{bmatrix} \epsilon & 0 & 0 \\ 0 & \epsilon & 0 \\ 1 & 0 & \epsilon \end{bmatrix}^{-1} \begin{bmatrix} 0 & 0 & -1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} \\ &= \begin{bmatrix} \epsilon^{-1} & 0 & 0 \\ 0 & \epsilon^{-1} & 0 \\ -\epsilon^{-2} & 0 & \epsilon^{-1} \end{bmatrix} \begin{bmatrix} 0 & 0 & -1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & -\epsilon^{-1} \end{bmatrix} \end{align} Again, it follows, that $\|G\|_\infty = \epsilon^{-1}$ and the iteration converges of all $f$ if and only if $\epsilon > 1$.
Your analysis does not eliminate the possibility that the iterations converge for some $\epsilon \leq 1$. Examing the iterations matrices removes this possibility.