M Theory Lesson 307

Such Hermitian matrices provide a basis for all $3 \times 3$ Hermitian $1$-circulants in the form Since the eigenvalues of each basis matrix take the values $(1,1,-1)$, the eigenvalues of the sum are of the form
$\lambda_1 = \alpha + \beta - \gamma$
$\lambda_2 = \alpha - \beta + \gamma$
$\lambda_3 = - \alpha + \beta + \gamma$
providing an alternative characterisation of mass triplets, replacing Brannen's $v$, $s$ and $\theta = 2/9$ with
$v = \alpha + \beta + \gamma$
$s = \frac{2 \gamma}{1.10265678}$
Note that when the Hermitian matrix is corrected by the appropriate phased diagonals, it arises as the Fourier transform of a circulant with only three non zero entries, representing the mass triplet. This Hermitian basis is only slightly different from the one derived by Brannen in his recent paper on generations.