Doubly Magic Matrices and the MNS

[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).

How can you characterize such matrices? The solution set for 2×2 matrices can be parameterized by any real number 0 <= a <= 1 as follows:

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?

The CKM matrix is usually not written with phases; the top right hand corner (i.e. the Cabibo corner) experimental data is roughly:

To write this as a doubly magic matrix, average the off diagonal values and solve for or therefore to obtain a = 0.951 and we find that the 2×2 magic version of the Cabibo mixing matrix is approximately:

Can the rest of the CKM matrix be put into doubly magic form? Word is that Kea, who started all this stuff, is working on the problem. Hopefully it will be interesting. Meanwhile, I need to write a post giving the weak hypercharge and weak isospin quantum numbers in 1-circulant and 2-circulant idempotent matrix form. The hope is that this will be a nice fit to the weak mixing angles in the same form, but I suspect that we will have to understand the quark mixing angles before this is all through.