Suppose A is a Hermitian positive definite matrix split into $\displaystyle A = C + C^{*} + D $ where $\displaystyle D$ is also Hermitian positive definite.

We show that $\displaystyle B=C+ \omega ^{-1} D$ is invertible. Consider the iteration $\displaystyle x_{n+1} = x_{n} + B^{-1} (b-Ax_{n})$ , with any initial iterate $\displaystyle x_{0}$ . Prove that $\displaystyle x_{n}$ converges to $\displaystyle x= A^{-1}b $ whenever $\displaystyle 0< \omega < 2$.

I suppose to show invertibility, we need to show $\displaystyle det(B) \neq 0$. But I am not sure how to show that. Also for the convergence, do we show $\displaystyle lim_{n \rightarrow \infty } \left\| x_{n}-x \right\|= 0 $ ? If yes, how?