State the Householder-John theorem and explain how it can be used in designing iterative methods for solving a system of linear equations $A \mathbf{x}=\mathbf{b}$. [You may quote other relevant theorems if needed.]

Consider the following iterative scheme for solving $A \mathbf{x}=\mathbf{b}$. Let $A=L+D+U$, where $D$ is the diagonal part of $A$, and $L$ and $U$ are the strictly lower and upper triangular parts of $A$, respectively. Then, with some starting vector $\mathbf{x}^{(0)}$, the scheme is as follows:

$$(D+\omega L) \mathbf{x}^{(k+1)}=[(1-\omega) D-\omega U] \mathbf{x}^{(k)}+\omega \mathbf{b} .$$

Prove that if $A$ is a symmetric positive definite matrix and $\omega \in(0,2)$, then, for any $\mathbf{x}^{(0)}$, the above iteration converges to the solution of the system $A \mathbf{x}=\mathbf{b}$.

Which method corresponds to the case $\omega=1 ?$