Past exam of the numerical linear algebra course of the University of Oxford / 2025 / 3 / a / iii / Solution

The claim is true, withIt is a symmetric positive-definite matrix: for every nonzero vector $z$, the fact that $A$ is an invertible matrix gives $z^{T}Mz=\Vert Az{\Vert }_2^2>0$. Since $A{x}_{\ast }=b$, every candidate $x$ satisfiesThe generalized minimal residual method therefore minimizes precisely this weighted norm of the error over the linear span of $Q_k$. The same matrix $M$ works for every $k$ and does not depend on $b$.