SOR method converges for $\left( \begin{array}{ccc}2& -1\\-2 & 2\end{array} \right)$

Question

Prove that the SOR method converges in $\mathbb{R}^n$ for the matrix $\left( \begin{array}{ccc}2& -1\\-2 & 2\end{array} \right)$ iff $\omega\in(0,2)$.

Answer 1

With the usual splitting of $A$ in the form $A=D+L+U$ to the diagonal, lower and upper triangular part, the iteration matrix of SOR is $$ T=(D+\omega L)^{-1}[(1-\omega)D-\omega U]. $$ Hence $$ T = \begin{bmatrix} 1 - \omega & \frac{\omega}{2} \\ \omega(1-\omega) & 1 - \omega + \frac{\omega^2}{2} \end{bmatrix}. $$ The characteristic polynomial of $T$ is $$ p(\lambda)=\lambda^2-\lambda\left(\frac{\omega^2}{2}-2\omega+2\right)+(\omega-1)^2. $$ I invite you to show that the roots $\lambda_+$ and $\lambda_-$ of $p$ satisfy $|\lambda_{\pm}|<1$ for $\omega\in(0,2)$.