Suppose $N\geq p$. Let $Q \in \mathbb{R}^{N \times p}$ has orthonormal columns $q_1,\dots,q_p$. Let $\Lambda \in \mathbb{R}^{N \times N}$ be a diagonal matrix with diagonal entries $\lambda_1 \geq \dots \geq \lambda_N \geq 0$.
What I am interested in is how to use the eigenvalues or singular values of $Q$ and $\Lambda$ to bound the eigenvalues of $Q^T \Lambda Q \in \mathbb{R}^{p \times p}$, especially the largest eigenvalue of $Q^T \Lambda Q$.
Let $f$ represent the linear function induced by your diagonal matrix, it has operator norm $\lambda_1$. You may think of $Q^T \Lambda Q$ as $i\circ w\circ f\circ q$ where $w$ is a projection, $i$ an isometric isomorphism (from the image of $w$ to some $p$-dimensional space) and $q$ is an isometry. Hence you may bound the eigenvalues of $Q^T \Lambda Q$ by $\lambda_1$. Also all the eigenvalues of the new matrix are non negative if you need a lower bound.