Chapter 10

Let \(\lambda \in \mathbb{C}\) be such that \(|\lambda|<1\) and define $$ J=\left[\begin{array}{ccccc} \lambda & 1 & 0 & \cdots & 0 \\ 0 & \lambda & \ddots & \ddots & \vdots \\ \vdots & \ddots & \ddots & 1 & 0 \\ \vdots & & \ddots & \lambda & 1 \\ 0 & \cdots & \cdots & 0 & \lambda \end{array}\right] $$ Prove that \(\lim _{k \rightarrow \infty} J^{k}=O\).

Prove that the Gauss-Seidel iteration converges whenever the matrix \(A\) is symmetric and positive definite.

Let \(A\) be a symmetric, positive definite, tridiagonal matrix. Prove that the SOR method converges for this matrix and for \(0<\omega<2\).

Let \(A\) be an \(m^{2} \times m^{2}\) matrix, which originates in the implementation of the five-point formula in a square \(m \times m\) grid. For every grid point \((r, s)\) we let $$ j_{(r, s)}:=m+s-r, \quad r, s,=1,2, \ldots, m $$ and we construct a vector \(j\) by assembling the components in the same order as that used in the matrix \(A\). Prove that \(j\) is an ordering vector of \(A\) and identify a permutation for which \(j\) is a compatible ordering vector.