Consider the linear system

$$A x=b$$

where $A \in \mathbb{R}^{n \times n}$ and $b, x \in \mathbb{R}^{n}$.

(i) Define the Jacobi iteration method with relaxation parameter $\omega$ for solving (1).

(ii) Assume that $A$ is a symmetric positive-definite matrix whose diagonal part $D$ is such that the matrix $2 D-A$ is also positive definite. Prove that the relaxed Jacobi iteration method always converges if the relaxation parameter $\omega$ is equal to $1 .$

(iii) Let $A$ be the tridiagonal matrix with diagonal elements $a_{i i}=\alpha$ and off-diagonal elements $a_{i+1, i}=a_{i, i+1}=\beta$, where $0<\beta<\frac{1}{2} \alpha$. For which values of $\omega$ (expressed in terms of $\alpha$ and $\beta$ ) does the relaxed Jacobi iteration method converge? What choice of $\omega$ gives the optimal convergence speed?

[You may quote without proof any relevant results about the convergence of iterative methods and about the eigenvalues of matrices.]