I'm reading this paper and came across this section, which I don't completely understand.
Equations (19) and (20) are "proportional to" equations, where $LHS = k RHS$.
Equation (22) is formed using the following logic
$a \propto b$ (19)
$c \propto d$ (20)
$\frac{b}{d} \propto \frac{a}{c}$
$da \propto bc$ (22)
If $da = bc$ then we could easily say $da - bc = 0$. However, in this case, we have something along the lines of
$da = k(bc)$
so
$da -k(bc) = 0$
but not necessarily
$da - bc = 0$.
Since we are building a set of constraints for a homogeneous linear system there must be strict equalities on the equations. However, I'm not sure how this can be justified.
By the way $H_j$ mentioned in the paper is a homography matrix which is scale invariant, as in $H_j \equiv kH_j$. My guess is that since $H_j$ is scale invariant, we can just set $||h_{j1}^s||^2 = k = 1$.