Triangular mass matrices of quarks and Cabibbo-Kobayashi-Maskawa mixing

Every nonsingular fermion mass matrix, by an appropriate unitary transformation of right-chiral fields, is equivalent to a triangular matrix. Using the freedom in choosing bases of right-chiral fields in the minimal standard model, reduction to triangular form allows for simple analytic expressions for the CKM matrix in terms of quark masses and a minimal set of parameters. Furthermore, diagonalization of the quark mass sectors can be shifted to one charge sector only, without losing the concise and economic triangular form. The corresponding effective triangular mass matrix is reconstructed, up to trivial phases, from the moduli of the CKM matrix elements, and vice versa, in a unique way. This reconstruction may also be relevant for the invariant measure of $\mathrm{CP}$ violation which we briefly discuss.