= Solution
For any nonzero <vector> $y\in\mathbb R^k$, <orthonormality> gives $\|Q_ky\|_2=\|y\|_2>0$. Since $A$ is a <symmetric positive-definite matrix>,
$$
\boxed{y^TT_ky=(Q_ky)^TA(Q_ky)>0}.
$$
The preceding part already gives symmetry, so $T_k$ is a <symmetric positive-definite matrix>. In fact the <spectral theorem> gives the useful bounds
$$
\lambda_{\min}(A)\|y\|_2^2
\leq y^TT_ky\leq\lambda_{\max}(A)\|y\|_2^2,
$$
which place its <eigenvalues> between the smallest and largest <eigenvalues> of $A$.
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