(a) State the Householder-John theorem and explain its relation to the convergence analysis of splitting methods for solving a system of linear equations $A x=b$ with a positive definite matrix $A$.

(b) Describe the Jacobi method for solving a system $A x=b$, and deduce from the above theorem that if $A$ is a symmetric positive definite tridiagonal matrix,

$$A=\left[\begin{array}{ccccc} a_{1} & c_{1} & & & \\ c_{1} & a_{2} & c_{2} & & \mathbf{0} \\ & \ddots & \ddots & \ddots & \\ \mathbf{0} & & c_{n-2} & a_{n-1} & c_{n-1} \\ & & & c_{n-1} & a_{n} \end{array}\right]$$

then the Jacobi method converges.

[Hint: At the last step, you may find it useful to consider two vectors $x=\left(x_{1}, x_{2}, \ldots, x_{n}\right)$ and $y=\left((-1) x_{1},(-1)^{2} x_{2}, \ldots,(-1)^{n} x_{n}\right)$.]