Let $D \in \mathbb{R}^{n \times n}$ be a diagonal matrix with non-negative elements, and let $Q \in \mathbb{R}^{n \times k}$, where $k What I have tried is
\begin{align*}
\mathbb{E}\left[ \left\| Q^TDQ \right\|_2\right] \leq \mathbb{E}\left[ \text{tr}\left( Q^TDQ \right)\right] = \text{tr}\left(\mathbb{E}\left[ Q^TDQ \right]\right),
\end{align*}
and then use
$\mathbb{E}\left[ Q^TDQ \right] = \frac{\| D\|_1}{n} I_k$, which is claimed by the OP here without proof: https://mathoverflow.net/questions/206485/eigenvalue-distribution-of-random-projection. This then gives the upper bound
$\mathbb{E}\left[ \left\| Q^TDQ \right\|_2\right] \leq \frac{k}{n} d_{max}$, where $d_{max}$ is the largest element of $D$. However, when I try this in Matlab, I find that $\left\| Q^TDQ \right\|_2$ is on average much larger than $\frac{k}{n} d_{max}$, which makes me think the equality from the link is not correct. What am I missing? If this is not correct, is there a better way to bound the original quantity? Upper bounding 2-norm by trace is probably quite loose, so I would be really interested to see if there is a tighter upper bound.
Spectral norm of a randomly projected matrix
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eigenvalues-eigenvectors
random-matrices