Quark mass matrices with four and five texture zeros, and the CKM matrix, in terms of mass eigenvalues

Using the triangular matrix techniques of Kuo, Mansour, and Wu and Chiu, Kuo, and Wu for the four and five texture zero cases, with vanishing (11) diagonal elements for U and D matrices, it is shown, from the general eigenvalue equations and hierarchy conditions, that the quark mass matrices and the Cabibbo-Kobayashi-Maskawa (CKM) matrix can be expressed (except for the phases) entirely in terms of quark masses. The matrix structures are then quite simple and transparent. We confirm their results for the five texture zero case but find, upon closer examination of all the CKM elements which our results provide, that six of their nine patterns for the four texture zero case are not compatible with experiments. In total, only one five texture zero and three four texture zero patterns are allowed.