  Structures play a significant role in the field of signal processing. As a
representative of structural data, low rank matrix along with its restricted
isometry property (RIP) has been an important research topic in compressive
signal processing. Subspace projection matrix is a kind of low rank matrix with
additional structure, which allows for further reduction of its intrinsic
dimension. This leaves room for improving its own RIP, which could work as the
foundation of compressed subspace projection matrix recovery. In this work, we
study the RIP of subspace projection matrix under random orthonormal
compression. Considering the fact that subspace projection matrices of $s$
dimensional subspaces in $\mathbb{R}^N$ form an $s(N-s)$ dimensional
submanifold in $\mathbb{R}^{N\times N}$, our main concern is transformed to the
stable embedding of such submanifold into $\mathbb{R}^{N\times N}$. The result
is that by $O(s(N-s)\log N)$ number of random measurements the RIP of subspace
projection matrix is guaranteed.
