Consider a random projection of some deterministic positive $m\times m$ Hermitian matrix $A$, defined as $B:=PUAU^\dagger P$, where the $m\times m$ unitary matrix $U$ is Haar random and $P$ is some fixed $m\times m$ projector with rank $n<m$. Does the spectrum of $B$ typically match approximately (up to normalisation) with the spectrum of $A$ ordered in decreasing order at the first $n$ eigenvalues?
(I'm not sure how to formulate the approximation precisely. Presumably I want the spectral distributions to be close as $n,m$ become large enough.)
Related post at mathoverflow, but it is not yet answered.
The answer is no. The largest eigenvalue of $PAP$ will match that of $A$ if and only if the eigenvector $x$ of $A$ associated with the largest eigenvalue of $A$ is an element of the range of $P$. However, for a given vector $x$, a "random" projection $P$ will contain $x$ with probability zero since the set of such projections is a proper Zariski closed subset of the set of projections.