Generalized two-angle parametrization of the Cabibbo-Kobayashi-Maskawa matrix

We demonstrate how to parametrize the Cabibbo-Kobayashi-Maskawa (CKM) matrix in terms of its eigenvalues and eigenvectors, generalizing a recent idea of Kielanowski's. In this version we are able to reproduce a symmetric CKM matrix with only two angles while predicting a range in the amount of $\mathrm{CP}$ violation. The relation between this parametrization and the standard one is studied. Some variations of this parametrization are worked out.