# Majorizing inequality on spectral norm of product of a random and a deterministic low-rank projection

Let $P$ be a rank $k$ uniformly randomly oriented projection matrix in ${\mathbb R}^d$ -- this is constructed as $R^T(RR^T)^{-1}R$ where $R$ is a $k\times d, k<d$ random matrix with i.i.d. 0-mean Gaussian entries with variance 1/d. Further, let T be a rank $r$ fixed deterministic projection matrix in ${\mathbb R}^d$, where $r\in\{1,...,d\}$.

Are the following true? (where $\lambda_{\max}(\cdot)$ denotes largest singular value)

• $Pr\{\lambda_{\max}(TP)> t\}\le Pr\{\lambda_{\max}(TR^TR)> t\}$

• $Pr\{\text{Trace}(TP)>t\}\le Pr\{\text{Trace}(TR^TR)> t\}$