From H&Z Multiple View Geometry
This ambiguity arises because it is possible to apply a projective transformation (rep- resented by a 4x4 matrix $H$) to each point $X_i$, and on the right of each camera matrix $P_j$ , without changing the projected image points, thus
$P_jX_i = (P_jH^{−1})(HX_i)$. (1.1)
There is no compelling reason to choose one set of points and camera matrices over the other. The choice of $H$ is essentially arbitrary, and we say that the reconstruction has a projective ambiguity, or is a projective reconstruction. However, the good news is that this is the worst that can happen. It is possible to reconstruct a set of points from two views, up to an unavoidable projective ambiguity. Well, to be able to say this, we need to make a few qualifications; there must be suffi- ciently many points, at least seven, and they must not lie in one of various well-defined critical configurations.
I'm confused about what exactly is a "projective ambiguity" and what 1.1 is trying to say.
I understand that
$P_jX_i = (P_jH^{−1})(HX_i) = P_j(H^{-1}H)X_i = P_jIX_i=P_jX_i$
However, what about this is ambiguous and why do we care that $P_iX_i = P_jIX_i$ ? More importantly, where's the loss of information? I wish I could ask more pointed questions but I think the purpose of this passage has flew over my head.
Your input data is $P_jX_i$: the images of the original points $X_i$ under the projections $P_j$. You'd like to reconstruct $X_i$ (and perhaps $P_j$ as well). But for any projective transformation $H$, there exists a set of points $HX_i$ which, when projected with projections $P_jH^{-1}$, leads to the same projected points. Which means that the projected points do not contain enough information to tell you which matrix $H$ (including the case of $H=I$, i.e. the case where you use no $H$ at all) corresponds to the real original situation. The reconstruction is only determined up to this one projective transformation of the world coordinates, thus it is ambiguous.
(Note that I haven't read the book, so I might be completely misunderstanding the context here, although I doubt it.)