I don't quite understand the Eqn(4) in GloVe paper.
$F({(w_i - w_j)}^T\tilde{w_k}) = \frac{P_ik}{P_jk}\:\:\:\:(3)$
$F({(w_i - w_j)}^T\tilde{w_k}) = \frac{F({w_i}^T\tilde{w_k})}{F({w_j}^T\tilde{w_k})}\:\:\:\:(4)$
I can understand that $F({(w_i - w_j)}^T\tilde{w_k}) = F({w_i}^T\tilde{w_k})-F({w_j}^T\tilde{w_k})$. But how it could be transformed into $F({w_i}^T\tilde{w_k})/F({w_j}^T\tilde{w_k})$(devision form). My understanding was that there's no devision in matrices calculus.
Or is it just the effect of F being group homomorphism b/w reals with addictive/multiplicative that the structure of devision on the right-hand side of Eqn(3) should match with the right-hand side of Eqn(4)?
*Editted(added the equation)