Let $A, B$ be two $m\times n$ matrices with $n > m$ and full row-rank. Define the following two (projection) matrices $$ \begin{align} P_A &= A^\top(AA^\top)^{-1}A \\ P_B &= B^\top(BB^\top)^{-1}B. \end{align} $$
Intuitively, if $A$ and $B$ are similar, does this mean $P_A$ and $P_B$ are similar?
For instance, let $\epsilon > 0$ be small, and assume $\|A - B\|_I < \epsilon$. Can I say $\|P_A - P_B\|_I < \epsilon$, or something along those lines?