### 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.

$\lambda_1 = \alpha + \beta - \gamma$

$\lambda_2 = \alpha - \beta + \gamma$

$\lambda_3 = - \alpha + \beta + \gamma$

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.

## 3 Comments:

hi can you post your all m theory lessons blogs please they are very useful

Anvesh, I will think about sorting things better, but it is tricky because Google have limited the Blogger options.

ok can u send them to my mail id anvesh.kadimi1@gmail.com

please thanks

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