natalia said:
I need to find the values of $\alpha$ for which the iteration converges.
Aha!
Actually, that turns out the same: you need $\alpha$ such that
$$||\alpha A || < 1$$
where $|| \cdot ||$ represents the matrix norm.
This is the same as:
$$|\alpha| \cdot || A || < 1$$
$$|\alpha| < \frac 1 {|| A ||}$$To explain, suppose the sequence converges (or should converge) to $y$.
Let $\Delta y^{(k)} = x^{(k)} - y$, which represents how far you are from the limit.
And let's call your matrix $A$ for ease of notation.
Then:
\begin{aligned}
x^{(k+1)} &= b + \alpha A x^{(k)} \\
&= b + \alpha A (y + \Delta y^{(k)}) \\
&= (b + \alpha A y) + \alpha A \Delta y^{(k)} \\
&= y + \alpha A \Delta y^{(k)}
\end{aligned}
That means that:
$$\Delta y^{(k+1)} = \alpha A \Delta y^{(k)}$$
In a worst case scenario this will only converge if
$$||\alpha A || < 1$$
where $|| \cdot ||$ represents the matrix norm.
