Is there any elementary way of showing that the Jacobi method converges for the following system?
$$\begin{pmatrix}10 & -2 & -11\\-3 & 10 & 2\\-1 & 1& 10\end{pmatrix}X= \begin{pmatrix} 13\\9\\11 \end{pmatrix}$$
The problem I face is that the iteration matrix has two complex eigenvalues, so how could I show that this eigenvalues have an absolute value which is less than $1$?
I think, if the complex eigenvalues are $\lambda_1=re^{\theta}$ and $\lambda_2=re^{-\theta}$, then you just need to consider $r$, together with the real eigenvalue.
What happens is that, as the iteration matrix $D^{-1}R$ gets multiplied by itself itteratively, its corresponding eigenvalue matrix is taken to higher powers and if $r$ and the real eigenvalue are less than $1$, they would go to $0$, eventually.
Set $D=diag(2,1,1)$, then $$ D^{-1}AD= \begin{pmatrix} 10&-1&-5.5\\ -6&10&2\\ -2&1&10 \end{pmatrix} $$ is diagonal-dominant. Which should be sufficient to show convergence of the Jacobi method.