# Daily Archives: October 9, 2008

## A New Parameterization for 3×3 Unitary Matrices

Given my final success in writing the CKM matrix as the sum of a real 1-circulant matrix and an imaginary 2-circulant matrix, the next step is to write down the parameterization that allows any 3×3 unitary matrix to be written in this elegant and natural form. Following the method used earlier to parameterize unitary 3×3 magic matrices, and correcting for a few typos (but the description of the method is correct).

We will use four real angles, $\theta, \alpha, \beta, \gamma$, as parameters. First, define 6 real numbers I, J, K, R, G, and B as follows:

The the following matrix is unitary:

This is the sum of a real 1-circulant matrix and an imaginary 2-circulant matrix. Note that the sum of the 3 elements of any row or column is equal to the complex phase $\exp(\pm i\gamma) = I+J+K \pm i(R+G+B)$ .