[Edit: For the doubly magic version of the CKM matrix, see CKM as a Magic Unitary Matrix. /Edit ] In the previous post, The MNS Matrix as Magic Square, we applied Kea’s observation that the CKM (quark weak mixing) matrix is approximately the sum of a 1-circulant and 2-circulant matrix to the MNS (lepton weak mixing) matrix. We found that the MNS matrix can also be written that way as well. The squared magnitudes (probabilities) of the MNS matrix are:
Messing around with the results of the previous post, we can obtain these values as the squared magnitudes of the sum of a rather simple 1-circulant and a 2-circulant matrix as follows:
The above matrix has the additional feature that its rows and columns all sum to . This number is a complex phase, that is, its squared magnitude is 1. So we can multiply by the complex conjugate of this and obtain an MNS matrix whose rows and columns sum to 1, which is the form given in the previous post, (other than I have swapped the columns around to match the literature).
So here’s the problem: Define a complex NxN matrix as “doubly magic” if its rows and columns all sum to unity, and its rows and columns give the same sum when their squared magnitudes are taken before summation. That is, a matrix M is doubly magic if:
for all k (and where the “[ ]” means absolute value, which I can’t fix on the computer I’m using tonight because the POS doesn’t have a modern paint).
The above matrix is fully characterized by its top left value. As a increases from 0 to 1, the top left value runs from 0 to (1 +- i)/2, and then on to 1. The squared magnitude is just a, so for each value of a other than 0 and 1, there are two solutions that are complex conjugates. Is there a way to generalize the above solution to 3×3 matrices?
For the case of the tribimaximal matrix, in deriving the doubly magic version of the matrix I found that there was only the given solution (plus the replacement i to -i). This is just like the above general solution for the 2×2 matrices. So, for an arbitrary 3×3 matrix of probabilities that sum to unity is there always a unique (up to complex conjugacy) doubly magic version of the amplitude? Is the MNS matrix just lucky or what? Does the fact that the MNS matrix can be written to be orthonormal play a part in this?