Assume solving the linear system $A \textbf x = \textbf b$, with an $A$ so large that nothing but iterative methods may be employed. Assuming $A$ induces a norm, I realized that it is often desired to minimize the residual in the norm induced by $A$ (and not, for instance, 2-norm).
Why is that the case? What is the advantage of using $A$-norm compared to 2-norm in different (general) optimization methods (GMRES, Gradient Decent/ Biconjugate Gradient decent etc.)?
PS: I'm more of a visual person and I appreciate it if you come up with a more graphical explanation. :)