For a question on Iterative Methods I have to show that the 2-norm of the residual is monotonically decreasing. We are given the following formula: $r^{(k+1)} = r^{(k)} - \alpha^{(k)} A z^{(k)}$ where $r^{(k)}$ is the residual of the iterative method at step $k$ and $\alpha$ the dynamic minimizer, given by $\alpha^{(k)} = \frac{(z^{(k)})^T r^{(k)}}{(z^{(k)})^T(Az^{(k)})}$, where $z^{(k)}$ is the preconditioned residual (although no specific preconditioner is given for this question).
What I've tried is to work out the 2-norm of the residual. I get this far:
$$||r^{(k+1)}||^2_2 = ||r^{(k)}||^2_2 + (\alpha^{(k)})^2 ||Az^{(k)}||^2_2 - 2\alpha^{(k)} ((r^{(k)})^T A z^{(k)}$$
My idea was to show that for the given value of $\alpha^{(k)}$, $(\alpha^{(k)})^2 ||Az^{(k)}||^2_2 - 2\alpha^{(k)} ((r^{(k)})^T A z^{(k)} \geq 0$ and then the inequality $||r^{(k+1)}||^2_2 \leq ||r^{(k)}||^2_2$ must hold, yet I am getting stuck at this point. Can anyone help me on the way to show this inequality?
