Hierarchical quark mass matrices

I define a set of conditions that the most general hierarchical Yukawa mass matrices have to satisfy so that the leading rotations in the diagonalization matrix are a pair of (2,3) and (1,2) rotations. In addition to Fritzsch structures, examples of such hierarchical structures include also matrices with (1,3) elements of the same order or even much larger than the (1,2) elements. Such matrices can be obtained in the framework of a flavor theory. To leading order, the values of the angle in the (2,3) plane ${(s}_{23})$ and the angle in the (1,2) plane ${(s}_{12})$ do not depend on the order in which they are taken when diagonalizing. We find that any of the Cabibbo-Kobayashi-Maskawa matrix parametrizations that consist of at least one (1,2) and one (2,3) rotation may be suitable. In the particular case when the $s_{13}$ diagonalization angles are sufficiently small compared to the product $s_{12}{s}_{23},$ two special CKM parametrizations emerge: the $R_{12}{R}_{23}{R}_{12}$ parametrization follows with $s_{23}$ taken before the $s_{12}$ rotation, and vice versa for the $R_{23}{R}_{12}{R}_{23}$ parametrization.