related by a small rotation which does not change the order of the elements if the departure of the original Kähler metric from the identity is also hierarchical as expected in models with a spontaneously broken family symmetry. Thus we still have $N_{ik}\leq \epsilon $ for $i\neq k$. Therefore, we would need $Y_{kj}>Y_{ij}$ for $i>k$, or more exactly the power in $\epsilon $ of $Y_{kj}$ is smaller than the power in $\epsilon $ of $Y_{ij}$ for $i>k$ so that $N_{ki}^{\ast }Y_{kj}>Y_{ij}$ is possible. However in this case we necessarily have $Y_{kj}^{t}=Y_{kj}>Y_{ij}+N_{ik}Y_{kj}=Y_{ij}^{t}$ for $i>k$ and $\ j\geq i$ (neglecting smaller contributions from $Y_{mj}$ with $m<k$ if present) and we arrive to an inconsistency with the initial statement $Y_{kj}^{t}<Y_{ij}^{t}$. Thus even with the weaker condition we need $Y_{kj}<Y_{ij}$ for $i>k$ and $\ j\geq i.$

So far we have discussed the transformations to canonical Kähler for the left handed fields. Now, we have to proceed exactly in the same way for the right-handed transformation. Clearly, if the Yukawa structures are also hierarchical we can perform the same analysis using upper triangular matrices and we would again arrive to the conclusion that corrections from the Kähler to any Yukawa element are always sub-dominant at 