Past exam of the numerical linear algebra course of the University of Oxford / 2025 / 3 / b / ii / Solution

For any nonzero vector $y\in {\mathbb{R}}^k$, orthonormality gives $\Vert Q_{k}y{\Vert }_2=\Vert y{\Vert }_2>0$. Since $A$ is a symmetric positive-definite matrix,The preceding part already gives symmetry, so $T_k$ is a symmetric positive-definite matrix. In fact the spectral theorem gives the useful boundswhich place its eigenvalues between the smallest and largest eigenvalues of $A$.