(a) Show that if $A$ and $B$ are real matrices such that both $A$ and $A-B-B^{T}$ are symmetric positive definite, then the spectral radius of $H=-(A-B)^{-1} B$ is strictly less than $1 .$

(b) Consider the Poisson equation $\nabla^{2} u=f$ (with zero Dirichlet boundary condition) on the unit square, where $f$ is some smooth function. Given $m \in \mathbb{N}$ and an equidistant grid on the unit square with stepsize $h=1 /(m+1)$, the standard five-point method is given by

$$u_{i-1, j}+u_{i+1, j}+u_{i, j-1}+u_{i, j+1}-4 u_{i, j}=h^{2} f_{i, j}, \quad i, j=1, \ldots, m$$

where $f_{i, j}=f(i h, j h)$ and $u_{0, j}=u_{m+1, j}=u_{i, 0}=u_{i, m+1}=0$. Equation $(*)$ can be written as a linear system $A x=b$, where $A \in \mathbb{R}^{m^{2} \times m^{2}}$ and $b \in \mathbb{R}^{m^{2}}$ both depend on the chosen ordering of the grid points.

Use the result in part (a) to show that the Gauss-Seidel method converges for the linear system $A x=b$ described above, regardless of the choice of ordering of the grid points.

[You may quote convergence results - based on the spectral radius of the iteration matrix - mentioned in the lecture notes.]