I'm taking the product of an orthogonal projection matrix (symmetric), $P$, with a diagonal matrix $D$ whose entries are either 0 or 1. So, $\tilde{P} = PD$ results in zeroing-out some of the columns of $P$.
I know that since both $P$ and $D$ have only 0s and 1s as eigenvalues, it must hold that $0 \leq \lambda_i \leq 1$ for any eigenvalue $\lambda_i$ of $\tilde{P}$ (as explained here: Eigenvalues of real diagonal matrix times orthogonal projection).
My question is: since we are essentially selecting a subset of the columns of $P$, does this impose a more restrictive upper bound on the eigenvalues? Specifically, I'm trying to show that they must be strictly less than 1.