# if A is symmetric positive definite the method JOR (over-relaxation) converges for a condition over $\omega$

Background:

A generalization of the Jacobi method for solving the system $$Ax = b$$ is the over-relaxation method (or JOR), which has the following iteration matrix:

$$B_{J_{\omega}} = \omega B_J + (1 - \omega) I$$

$$B_J = I - D^{-1}A$$ is the iteration matrix for the Jacobi method (the matrix $$D$$ is the diagonal of $$A$$).

I know a iterative method converges if, and only if, its iteration matrix $$B$$ spectral radius is less than $$1$$, i.e. $$\rho(B) < 1$$.

I want to prove:

If $$A$$ is symmetric positive definite, then the JOR method is convergent if $$0 < \omega < \dfrac{2}{\rho(D^{-1}A)}$$

This question is the theorem 4.4 on the book "Numerical Mathematics", by Alfio Quarteroni - second edition. The book says the result is "immediate" from the information I wrote above, but I can't see it.

If $$A$$ is SPD, then $$D^{-1}A$$ is diagonalizable and has positive eigenvalues in the interval $$(0,\rho]$$ with $$\rho:=\rho(D^{-1}A)$$. Let $$\lambda$$ and $$x$$ be an eigen-pair of $$D^{-1}A$$. Then $$B_{J_\omega}x=[\omega B_J+(1-\omega)I]x=[\omega (I-D^{-1}A)+(1-\omega)I]x =(1-\lambda\omega)x,$$ so $$\mu:=1-\lambda\omega$$ and $$x$$ is an eigen-pair of $$B_{J_\omega}$$.
We need that $$-1<\mu<1$$, which (since $$0<\lambda$$) is equivalent to $$0<\omega<\frac{2}{\lambda}.$$ So to confine $$\mu$$ in this interval for all $$\lambda$$ in $$(0,\rho]$$, this leads to the condition $$0<\omega<\frac{2}{\rho}.$$