You want to show that is invertible, but appears in the expression which you plan on using to do that... Are you sure you copied the problem properly?
Suppose A is a Hermitian positive definite matrix split into where is also Hermitian positive definite.
We show that is invertible. Consider the iteration , with any initial iterate . Prove that converges to whenever .
I suppose to show invertibility, we need to show . But I am not sure how to show that. Also for the convergence, do we show ? If yes, how?