Given a 3-vector of complex numbers, (A,B,C), define its discrete Fourier transform as
where . That is, I’ll use lower case letters to denote the discrete Fourier transforms of UPPER case letters. The above leaves off a factor of but it will do.
Of interest today will be vectors (A,B,C) which happen to satisfy A+B+C = 0. These are eigenvectors of the Democratic D matrix
that is, the matrix all of whose entries are equal to the complex number D. Of course their eigenvalues are zero. None of this is particularly interesting until we move from linearity to bilinearity and work with the discrete Fourier transforms of 3×3 matrices.
Define the Fourier transform of a 3×3 matrix U as where is the matrix:
where . With this definition, the discrete Fourier transform of the democratic matrix D, is:
This is a nice simplification.
Now let A+B+C=0 and compute some discrete Fourier transforms of four kinds of matrices, 1-circulant, 2-circulant, and two new types I will call “bra” and “ket” for obvious reasons. Untransformed matrices on the left, their transforms on the right, note that they fit together like the pieces of a jigsaw puzzle:
The discrete Fourier transform for 3×3 matrices is 1-1 and onto, so the above gives a proof that we can always write a 3×3 matrix uniquely as the sum of a democratic, 1-circulant, 2-circulant, bra, and a ket matrix, each of which, other than the democratic, is subject to the constraint that A+B+C = 0. This gives a decomposition of a 3×3 matrix into five parts, with the democratic part giving one and the others two each degrees of freedom, for a total of nine (complex) degrees of freedom.
Some time ago, I gave a parameterization that seems to be a complete parameterization of the magic unitary 3×3 matrices. This uses 4 real parameters. We can add five more parameters by multiplying rows and columns by arbitrary complex phases. This gives a total of nine parameters, just what are needed for an arbitrary unitary matrix. So I’m wondering how to prove that this parameterization is complete or not.
Of the five components of a 3×3 matrix as given above, three are magic, that is, three have rows and columns sum to the same value. The bra and ket forms have rows or columns that sum to the same value but not both. Consequently, we know that any magic matrix can be written uniquely as the sum of a democratic, a 1-circulant, and a 2-circulant and this must also apply to the magic unitary matrices. Because the decomposition is 1-1 and onto, we know that the magic matrices have 1+2+2 = 5 complex degrees of freedom.
The 1-circulant, 2-circulant, and democratic matrices are closed under multiplication and addition, thus the magic matrices are a subalgebra in the algebra of 3×3 matrices. The unitary matrices are also a subalgebra of the 3×3 matrices and so are the 3×3 magic unitary matrices. Now the 4 parameter parameterization I gave for them was under the additional restriction that the sum of the rows and columns, in addition to being equal, was required to be equal to 1. We can generalize with a 5th parameter to multiply the whole matrix with a complex phase and therefore make the sum of each row and column be an arbitrary complex phase. And this gives 5 parameters that define the magic unitary matrices completely.
This is kind of funny. Magic 3×3 matrices have five complex degrees of freedom, while magic 3×3 unitary matrices have five real degrees of freedom. Could there be a relationship between them?
All this discussion has to do with the democratic, 1-circulant, and 2-circulant portions of the matrices. What about the bra and ket portions?
The method I’ve been using to get a general unitary matrix into magic form is by multiplying the rows and columns by arbitrary complex constants. Ignoring the overall complex phase, we get to pick two rows and two columns for this treatment:
Comparing with the table of discrete Fourier transforms, the matrices of complex phases in the above equation can be seen to be the discrete Fourier transforms of 1-circulant matrices, that is, they are of the form
where, as before, the complex numbers A,B, and C are related by A+B+C = 0. This gives two complex degrees of freedom or four real degrees of freedom, again just twice what we need to arrange for multiplication by two complex phases.
So I think I’m getting fairly close to cobbling together a proof that the new parameterization does cover all the unitary matrices, when it is extended by multiplying columns and rows by complex phases.
The funny thing about all this is that it has the feeling of complete triviality, but I’ve never seen the unitary matrices parameterized so nicely.