# Invertible matrix

1. Apr 22, 2010

### math8

Suppose A is a Hermitian positive definite matrix split into $$A = C + C^{*} + D$$ where $$D$$ is also Hermitian positive definite.
We show that $$B=C+ \omega ^{-1} D$$ is invertible. Consider the iteration $$x_{n+1} = x_{n} + B^{-1} (b-Ax_{n})$$ , with any initial iterate $$x_{0}$$ . Prove that $$x_{n}$$ converges to $$x= A^{-1}b$$ whenever $$0< \omega < 2$$.

I suppose to show invertibility, we need to show $$det(B) \neq 0$$. But I am not sure how to continue.