I need to solve this problem:
Let $A\in \mathcal M_n(\mathbb R)$ a positive definite matrix, $B \in \mathbb R^n $, and $AX=B$ a system of linear equations. We use an iterative method which generates a sequence $ \{X_k\}_{n\in \mathbb N}$ to approximate the value of the solution $\bar X$ . We consider the residual vector $R_k = B - AX_k$ and the error $E_k = X_k - \bar X$.
a) Prove that $ \left<R_k, E_k \right> \le 0 \ \forall k $
b) Prove that if there's a $k_0$ so that $ \left<R_{k_0}, E_{k_0} \right> = 0$ then $X_{k_0} = \bar X$
I think this problem must be easy but somehow I am stuck. So here is what I have done so far:
$ \left<R_k, E_k \right> = \left< B - AX_k, X_k - \bar X \right>=\left< B, X_k - \bar X \right> - \left<AX_k, X_k - \bar X \right> =\left< B, X_k \right> - \left< B, \bar X \right> - \left<AX_k, X_k\right> +\left< AX_k, \bar X \right> =\left< A \bar X, X_k \right> - \left< A \bar X, \bar X \right> - \left<AX_k, X_k\right> +\left< AX_k, \bar X \right> $
Where I have used that $B= A\bar X $
Now I know that since A is a definite positive matrix $\left< A \bar X, \bar X \right> >0 $ and $\left< AX_k, X_k \right> > 0$.
Also, since A is symmetric I think that $ \left< A \bar X, X_k \right> = \left< AX_k, \bar X \right> $
However I don't know how to prove a) nor b).
Thank you for your help !!