Suppose a random projection $P$ in $\mathbb{R}^d$ onto a random n-dimensional subspace in $\mathbb{R}^d$ uniformly distributed in the Grassmannian $G_{d, n}$ (the projection of the row space of a random matrix $X \in \mathbb{R}^{n \times d}$ where each entry is i.i.d sampled from $\mathcal{N}(0, 1)$). Is $I - P$ a random projection onto a random (d-n)-dimensional subspace uniformly distributed in Grassmannian $G_{d, d-n}$?
I'm interested in applying lemma 5.3.2. from Roman Vershynin's book High-dimensional probability (download), with the respect to the projection to the orthogonal row space of $X$ as described above.
Yes. The null space of $P$ is uniformly distributed on $G_{d, d-n}$. But this is equal to the range space of $I-P$.
To show that the null space is uniformly distributed on $G_{d, d-n}$, we first define the uniform measure $\gamma_n$ on $G_{d, n}$. Consider Haar measure $\theta$ on $O(n)$. Fix $V$ in $G_{d, n}$. For a subset $A$ of $G_{d, n}$ define $\gamma_n(A) = \theta(\{g\in O(n) | gV\in A\})$.
Let $A$ be a subset of $G_{d, n}$, and let $B = \{W^{\perp} | W \in A\}$. Note that if $A$ corresponds to the projections of $P$ in $\mathbb{R}^d$, then $B$ corresponds to their null spaces. Then \begin{align} \gamma_{d-n}(B) &= \theta(\{g\in O(n) | gV^{\perp}\in B\}) \\ &= \theta(\{g\in O(n) | gV\in A\}) \\ &= \gamma_{n}(A). \end{align}