Question: Let's assume $A$ is a nonsingular matrix and we will use the iteration $$x+δr \to x$$ to solve for $Ax=b$, where $r=b-Ax$. We choose $δ$ in order to minimize the function $\phi(x)={\|Ax-b\|}_2^2$ where $x$ has vectors of the form $x+δr$. Justify whether the iterative method would converge.
Confusion: The approach i thinking is to find a $r\ne0$ s.t. $d=0$. But I do not really understand the minimization function $\phi(x)={\|Ax-b\|}_2^2$ and how would i find a $r=0$ to minimize it? Or is it better to find the standard iterative matrix first? Thanks in advance.
Put all in one formula and see what happens $$ \|Ax_+-b\|^2=\|A(x+\delta r)-b\|^2=\|r+δAr\|^2=\|r\|^2+2δ\langle r,\,Ar\rangle+δ^2\|Ar\|^2 $$ The minimum of this quadratic polynomial happens for $δ=-\dfrac{⟨r,Ar⟩}{∥Ar∥^2}$, the value of the quadratic there is $\|r\|^2-\dfrac{⟨r,Ar⟩^2}{∥Ar∥^2}$. So you want that $⟨r,Ar⟩=0$, $r\perp Ar$ for some $r=Ax-b\ne 0$.
As $2⟨r,Ar⟩=⟨r,(A+A^\top)r⟩$, this would be possible if the symmetric part of $A$ has a non-trivial kernel.