Chapter 12

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 \(\lim _{k \rightarrow \infty} J^{k}=O .\)

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

Demonstrate that $$ \sum_{\ell=1}^{d} \sin ^{2}\left(\frac{\pi j \ell}{d+1}\right)=\frac{1}{2}(d+1), \quad j=1,2, \ldots, d $$ thereby verifying (12.29).