# 3.II.16B

(a) Consider a system of linear equations $A x=b$ with a non-singular square $n \times n$ matrix $A$. To determine its solution $x=x^{*}$ we apply the iterative method

$x^{k+1}=H x^{k}+v .$

Here $v \in \mathbb{R}^{n}$, while the matrix $H \in \mathbb{R}^{n \times n}$ is such that $x^{*}=H x^{*}+v$ implies $A x^{*}=b$. The initial vector $x^{0} \in \mathbb{R}^{n}$ is arbitrary. Prove that, if the matrix $H$ possesses $n$ linearly independent eigenvectors $w_{1}, \ldots, w_{n}$ whose corresponding eigenvalues $\lambda_{1}, \ldots, \lambda_{n}$ satisfy $\max _{i}\left|\lambda_{i}\right|<1$, then the method converges for any choice of $x^{0}$, i.e. $x^{k} \rightarrow x^{*}$ as $k \rightarrow \infty$.

(b) Describe the Jacobi iteration method for solving $A x=b$. Show directly from the definition of the method that, if the matrix $A$ is strictly diagonally dominant by rows, i.e.

$\left|a_{i i}\right|^{-1} \sum_{j=1, j \neq i}^{n}\left|a_{i j}\right| \leq \gamma<1, \quad i=1, \ldots, n,$

then the method converges.