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

Write $x=Q_{k}y$ and $\beta =\Vert b{\Vert }_2$. The Arnoldi decomposition and $b=\beta Q_{k+1}{e}_1$ giveBecause $Q_{k+1}$ has orthonormal columns, it preserves the Euclidean norm. The generalized minimal residual method therefore reduces to the small linear least-squares problemUse Givens rotations on the upper Hessenberg matrix to obtain a QR decompositionwhere $Z$ is an orthogonal matrix and $R_k$ is an upper triangular matrix. Then solve $R_k{y}_k=g_{1:k}$ by backward substitution in a triangular system and form $Q_k{y}_k$. The resulting minimum linear-system residual has Euclidean norm $|g_{k+1}|$.

The small problem has a unique answer: if ${\tilde{H}}_{k}y=0$, the Arnoldi decomposition gives $A{Q}_{k}y=0$. The invertible matrix $A$ implies $Q_{k}y=0$, and the orthonormality of $Q_k$ then gives $y=0$. Thus ${\tilde{H}}_k$ has full column rank and $R_k$ is nonsingular.