(a) Describe the Jacobi method for solving a system of linear equations $A \boldsymbol{x}=\boldsymbol{b}$ as a particular case of splitting, and state the criterion for its convergence in terms of the iteration matrix.

(b) For the case when

$$A=\left[\begin{array}{lll} 1 & \alpha & \alpha \\ \alpha & 1 & \alpha \\ \alpha & \alpha & 1 \end{array}\right]$$

find the exact range of the parameter $\alpha$ for which the Jacobi method converges.

(c) State the Householder-John theorem and deduce that the Jacobi method converges if $A$ is a symmetric positive-definite tridiagonal matrix.