The following is classical in numerical linear algebra.

Let $A\in M_n({\mathbb R})$ be tridiagonal (one) with an invertible diagonal D (two). Assume that the eigenvalues of $J:=I_n-D^{-1}A$ belong to $(-1,1)$ (my three). Then the relaxation method converges for every choice of the relaxation parameter $\omega$ in the interval $(0,2)$; the optimal parameter is unique and equal to $$\omega^*=\frac{2}{1+\sqrt{1-\rho(J)^2}}.$$

If you do not like my two, you can take $\omega\in(0,2)$ as an hypothesis.