Let $A\in\mathbb{R}^{n\times m}$ with $n\geq m$ have full rank and let $b\in\mathbb{R}^{n}$ be nonzero. Then $H=A^{T}A\in\mathbb{R}^{m\times m}$ is symmetric and positive definite and for $k\leq m$ and $g=A^{T}b\in\mathbb{R}^{m}$ we can consider the matrix $M_{k}=[g,Hg,\ldots,H^{k-1}g]\in\mathbb{R}^{m\times k}$ which spans a (at most) $k$-dimensional subspace of $\mathbb{R}^{m}$ (the Krylov-space $\mathcal{K}_{k}(H,g)$). Furthermore, let $Q_{k}R_{k}=M_{k}$ be a QR decomposition. I am now interested in the properties of the positive definite and symmetric matrix $Q_{k}^{T}H^{2}Q_{k}\in\mathbb{R}^{k\times k}$. From the orthonormality of $Q_{k}$ it follows that the matrix is pentadiagonal.
The question is: Can the element $Q_{k}^{T}Hg\in\mathbb{R}^{k}$ be in the span of the last $k-1$ columns of $Q_{k}^{T}H^{2}Q_{k}\in\mathbb{R}^{k\times k}$? This is equivalent to the question whether the first component of $y_{k}=[Q_{k}^{T}H^{2}Q_{k}]^{-1} Q_{k}^{T}Hg$ can vanish. Can any of the components of $y_{k}$ vanish?