I am stuck on a finding a probablistic bound on a nonstandard random matrix. I looked around on the internet and couldn't find any results. This could be because I don't know the key words or because results are sparse. Any pointers from readers here would be of great help.
Let $n>k$. Let $R$ be a uniformly orthonormal real $n \times k$ matrix, i.e., $R^TR = I$ and $R$ is uniformly distributed on the manifold of such real matrices. Suppose $M$ is some deterministic real matrix of compatible dimensions. Note that $RR^T$ is a singular matrix.
I am looking for a result which says $\|RR^TM\| \leq f(n,k,\delta)$ with probability at least $1-\delta$ for some $\delta \in (0,1)$ and a function $f$. Since $R$ is orthonormal, there is a trivial bound $\|RR^TM\| \leq \|RR^T\| \|M\|=\|M\|$ (since any orthonormal matrix $R$ satisfies $\|RR^T\|=1$). I think one ought to get a better bound by invoking the randomness in $R$.
Prima facie, it appears that this ought to be related to the distribution of singular Wishart matrices. But I am unable to make this precise. I would appreciate any help in this matter.