Daily Archives: August 12, 2008

Unitary Circulant Matrices and Idempotentcy

Let A_k be a complete set of N annihilating primitive idempotents. That is, \Sigma_k A_k = 1 , and A_jA_k = 0 when j is not k, and A_k A_k = A_k for all k. Also suppose that each A_k is Hermitian.

Let \alpha_k be a set of N real numbers. Then:
\Sigma_k e^{i\alpha_k}A_k is unitary.
I’m guessing that the reader will find the proof immediately. If not, ask in the comments and I’ll give the short proof.

I’ve been working on the CKM matrix recently, which is a 3×3 unitary matrix. For 3×3 matrices, the simplest complete set of annihilating primitive idempotents is the diagonal primitive idempotents, that is, the matrices that are zero except for a single one somewhere on the diagonal:
Diagonal 3x3 matrix complete set of annihilating primitive idempotents
The unitary matrices generated by this set are simply the diagonal matrices with complex phases down the diagonal.

The 1-Circulant Primitive Idempotents

Around here, our favorite complete set of mutually annihilating primitive idempotents for the 3×3 matrices are the 1-circulant ones:
Complete set of annihilating circulant primitive idempotent 3x3 matrices
Label the above three 3×3 matrices as P_I, P_J, P_K .

Interpreted as density matrix states, these are generated from the bra / ket states as shown in the right side of the above. Just as all other pure density matrix states created from state vectors, they are Hermitian. In addition, they are all magic. The rows and columns of P_I sum to unity, while the rows and columns of the P_J and P_K sum to zero.

So these are just what we need to write down an elegant parameterization of the 1-circulant unitary matrices:
e^{i\theta_I}P_I + e^{i\theta_J}P_J + e^{i\theta_K}P_K.
This is a unitary matrix whose rows and columns sum to e^{i\theta_I} .

The previous post showed a parameterization of the magic unitary 3×3 matrices. These included 2-circulant parts as well as 1-circulant. In the above, the P_I primitive idempotent is both 1-circulant and 2-circulant, while the other two matrices are 2-circulant. Can we get an elegant parameterization of the magic matrices this way?
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