Koide Review III

So the set of nine unscaled Koide eigenvalues for the lepton masses may now be expressed as the matrix product where the left hand factor is the tribimaximal mixing matrix. The right hand factor does not look like a probability matrix, but we could take a fixed constant off all the ${\lambda }_{ij}$ so that this factor, and also the tribimaximal matrix, take the form The two factors use the values $(\delta ,\theta )=$ $(0,\omega )$ and $(2/9,\pi /12)$ respectively. Similarly, the democratic matrix offset uses $(\delta ,\theta )=$ $(0,\pi /4)$. Note that we need the $\sqrt{2}$ coefficients to make magic matrices, when the angles are right.