form for $N$ and taking into account that the hierarchy in the left handed angles implies that $q_j > q_i$, we have that $N_{i < j} = (\theta^*/M)^{(q_j - q_i)}= \epsilon^{(q_j - q_i)}$. This form is forced from the requirement of invariance under the Abelian symmetry of the canonically normalised Yukawa element. In the case of non-Abelian symmetries the hierarchy in the Yukawa matrices is obtained from the smallness of vevs of the different flavon fields. In principle, only the vev defining the third generation can be ${\mathcal{O}}(1)$ while vevs in the direction of the second or first generation are $\leq \epsilon$. Given that the flavour structure of the Kähler matrices is necessarily generated in terms of the same flavon vevs, we have that any off-diagonal term in the Kähler matrices involves at least one power of the small vevs and hence $N_{i < j} \leq \epsilon$. A similar argument applies to the up quark $RH$ sector, $N_{ii}^{R,u}\simeq 1$ and $N_{ik}^{R,u}\leq \varepsilon$ but in the down quark $RH$ sector the expansion parameter must be the larger one, $\bar{\varepsilon}$, so $N_{ii}^{R,d}\simeq 1$ and $N_{ik}^{R,d}\leq \bar{\varepsilon}.$

In fact it is easy to prove that for the hierarchical textures of interest here the 