The Poisson equation on the unit square, equipped with zero boundary conditions, is discretized with the 9-point scheme:

$$\begin{aligned} &-\frac{10}{3} u_{i, j}+\frac{2}{3}\left(u_{i+1, j}+u_{i-1, j}+u_{i, j+1}+u_{i, j-1}\right) \\ &+\frac{1}{6}\left(u_{i+1, j+1}+u_{i+1, j-1}+u_{i-1, j+1}+u_{i-1, j-1}\right)=h^{2} f_{i, j} \end{aligned}$$

where $1 \leqslant i, j \leqslant m, u_{i, j} \approx u(i h, j h)$, and $(i h, j h)$ are the grid points with $h=\frac{1}{m+1}$. We also assume that $u_{0, j}=u_{i, 0}=u_{m+1, j}=u_{i, m+1}=0$.

(a) Prove that all $m \times m$ tridiagonal symmetric Toeplitz (TST-) matrices

$$H=[\beta, \alpha, \beta]:=\left[\begin{array}{cccc} \alpha & \beta & & \\ \beta & \alpha & \ddots & \\ & \ddots & \ddots & \beta \\ & & \beta & \alpha \end{array}\right] \in \mathbb{R}^{m \times m}$$

share the same eigenvectors $\boldsymbol{q}_{k}$ with the components $(\sin j k \pi h)_{j=1}^{m}$ for $k=1, \ldots, m$. Find expressions for the corresponding eigenvalues $\lambda_{k}$ for $k=1, \ldots, m$. Deduce that $H=Q D Q^{-1}$, where $D=\operatorname{diag}\left\{\lambda_{k}\right\}$ and $Q$ is the matrix $(\sin i j \pi h)_{i, j=1}^{m} .$

(b) Show that, by arranging the grid points ( $i h, j h)$ into a one-dimensional array by columns, the 9 -points scheme results in the following system of linear equations of the form

$$A \boldsymbol{u}=\boldsymbol{b} \Leftrightarrow\left[\begin{array}{cccc} B & C & & \\ C & B & \ddots & \\ & \ddots & \ddots & C \\ & & C & B \end{array}\right]\left[\begin{array}{c} \boldsymbol{u}_{1} \\ \boldsymbol{u}_{2} \\ \vdots \\ \boldsymbol{u}_{m} \end{array}\right]=\left[\begin{array}{c} \boldsymbol{b}_{1} \\ \boldsymbol{b}_{2} \\ \vdots \\ \boldsymbol{b}_{m} \end{array}\right]$$

where $A \in \mathbb{R}^{m^{2} \times m^{2}}$, the vectors $\boldsymbol{u}_{k}, \boldsymbol{b}_{k} \in \mathbb{R}^{m}$ are portions of $\boldsymbol{u}, \boldsymbol{b} \in \mathbb{R}^{m^{2}}$, respectively, and $B, C$ are $m \times m$ TST-matrices whose elements you should determine.

(c) Using $\boldsymbol{v}_{k}=Q^{-1} \boldsymbol{u}_{k}, \boldsymbol{c}_{k}=Q^{-1} \boldsymbol{b}_{k}$, show that (2) is equivalent to

$$\left[\begin{array}{cccc} D & E & & \\ E & D & \ddots & \\ & \ddots & \ddots & E \\ & & E & D \end{array}\right]\left[\begin{array}{c} \boldsymbol{v}_{1} \\ \boldsymbol{v}_{2} \\ \vdots \\ \boldsymbol{v}_{m} \end{array}\right]=\left[\begin{array}{c} \boldsymbol{c}_{1} \\ \boldsymbol{c}_{2} \\ \vdots \\ \boldsymbol{c}_{m} \end{array}\right]$$

where $D$ and $E$ are diagonal matrices.

(d) Show that, by appropriate reordering of the grid, the system (3) is reduced to $m$ uncoupled $m \times m$ systems of the form

$$\Lambda_{k} \widehat{v}_{k}=\widehat{c}_{k}, \quad k=1, \ldots, m$$

Determine the elements of the matrices $\Lambda_{k}$.