Suppose a system $$Ax=b$$ is given, with $A\in\mathbb{R}^{n\times n}$ being a symmetric positive-definite matrix, and some non-zero $b\in\mathbb{R}^n$. The gradient method with optimum step length can be written as $$x_{k+1}=x_k-\alpha_k\cdot g_k,$$ with $g_k=Ax_k-b$ and $\alpha_k=\frac{g_k^Tg_k}{g_k^TAg_k}$.
Can it be proven that the above iterative procedure converges to $\bar{x}=A^{-1}b$, regardless of the initialization?
Can the above iterative scheme be regarded as a linear fixed-point iteration? In fact, what is the precise meaning of linear in the term linear fixed-point iteration?