M Theory Lesson 257

Unitary magic matrices with non-negative rational entries, such as the norm square of the neutrino mixing matrix, form a semigroup because the product of two such matrices results in another matrix of the same kind. Restricting to 1-circulant unitary magic matrices results in a smaller semigroup, since products of 1-circulants are again 1-circulants. Observe that in a product of the form the difference between the two entries in the resulting circulant is $(a - b)(d - c)$, namely the product of the differences in the components. In particular, the power $M^{n}$ of a single such 1-circulant $M$ results in a difference of $(a - b)^{n}$, which cannot be zero for finite $n$ if $a \neq b$. So the only way such a power can result in the democratic unitary magic matrix $D = (1/3,1/3,1/3)$ is if it is an infinite power. Moreover, since $a, b < 1$, it is always the case that an infinite power will converge to $D$, that is $M^{\infty} = D$.