(i) The five-point equations, which are obtained when the Poisson equation $\nabla^{2} u=f$ (with Dirichlet boundary conditions) is discretized in a square, are

$$-u_{m-1, n}-u_{m, n-1}-u_{m+1, n}-u_{m, n+1}+4 u_{m, n}=f_{m, n}, \quad m, n=1,2, \ldots, M,$$

where $u_{0, n}, u_{M+1, n}, u_{m, 0}, u_{m, M+1}=0$ for all $m, n=1,2, \ldots, M$.

Formulate the Gauss-Seidel method for the above linear system and prove its convergence. In the proof you should carefully state any theorems you use. [You may use Part (ii) of this question.]

(ii) By arranging the two-dimensional arrays $\left\{u_{m, n}\right\}_{m, n=1, \ldots, M}$ and $\left\{b_{m, n}\right\}_{m, n=1, \ldots, M}$ into the column vectors $\mathbf{u} \in \mathbb{R}^{M^{2}}$ and $\mathbf{b} \in \mathbb{R}^{M^{2}}$ respectively, the linear system described in Part (i) takes the matrix form $A \mathbf{u}=\mathbf{b}$. Prove that, regardless of the ordering of the points on the grid, the matrix $A$ is symmetric and positive definite.