Let be a complete set of N annihilating primitive idempotents. That is, , and when j is not k, and for all k. Also suppose that each is Hermitian.

Let be a set of N real numbers. Then:

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:

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:

Label the above three 3×3 matrices as .

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 sum to unity, while the rows and columns of the and sum to zero.

So these are just what we need to write down an elegant parameterization of the 1-circulant unitary matrices:

This is a unitary matrix whose rows and columns sum to .

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 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?

If we could add two 2-circulant matrices to our list of mutually annihilating primitive idempotents, the result would be nicely regular. Of course this is impossible. There would be two matrices that give the 1-circulant part, that is , two new matrices for the 2-circulant part, and the 5th matrix gives a parameter that is both 1-circulant and 2-circulant (and whose parameter would be set to zero if we wish to arrange for the matrix to have rows and columns summing to 1).

The and matrices are are related to the I, J and K permutation matrices by a cyclic sort of Fourier transform:

It’s natural to see what happens when we do the same thing to the R, G, and B permutation matrices.

As with the I, J, K Fourier transform, we have to choose an arbitrary one of the {R, G, B} as the multiplicative constant, the equivalent of I. Let’s go with R. Then define:

Since , we don’t have to deal with that matrix. The other two are:

The above two matrices are not Hermitian, but they are each other’s Hermitian conjugate. In addition, instead of being idempotent, they are nilpotent; they square to zero: . And they do not annihilate each other. Instead, they multiply to give 1-circulant primitive idempotents:

These last two equations are similar to the relations among the permutations: GB = K, BG = J. But they do annihilate against :

Finally, they either annihilate or are left unchanged on multiplication by :

So. Can I cobble this together into another parameterization of the magic unitary 3×3 matrices?

I think it probably will work. Maybe I’ll work it out later tonight and edit it into the bottom of this post.

**Some Magic in the Elementary Particles Literature**

Since writing the previous blog post, I found a literature reference to magic matrices in the generation structure of the elementary particles: Magic Neutrino Mass Matrix and the Bjorken-Harrison-Scott Parameterization , C. S. Lam, Phys. Lett. B640 (2006) 260-262. Also see Status of Tri/Bi-Maximal Neutrino Mixing, P. F. Harrison, W. G. Scott, Second Workshop on Neutrino Oscillations in Venice: “whereby the neutrino mass matrix in the lepton flavour basis takes the form of a general S3 group matrix (3 x 3 `magic-square’).”

Hi Carl,

In reading the wikipedia entry on Idempotent, I noticed that idempotence may occur in Boolean algebra and projections of linear algebra. Not listed is MAX Plus or tropical algebras.

In reading Michiel Hazewinkel, ‘encyclopaedia of mathematics’, p 314-317, idempotence is discussed with Maslov spaces and a correspondence principle with respect to Legendre and Fourier transforms, the Hamilton-Jacobi theorem with the Bellman equation and as probability measures for Markov stochastic processes in decision problems.

If I am interpreting the latter reference correctly, the correspondence principle seems to indicate that nilpotents can be transformed into idempotents, something like “Bohr’s correspondence principle in quantum mechanics” [p 315].

Yes, there is a close relationship between idempotents, nilpotents, anb the “mutually unbiased bases” (MUBs) that get discussed around here.

I work in Clifford algebras, where the idempotents are defined by “commuting roots of unity”. That would be a set of a_n that all square to +1, and commute, and that are linearly indpendent (i.e. can’t include +b and -b). if such a set has as many elements as possible, say N, then products of the form:

A = (1 +- a_1)/2 (1 +- a_2)/2 … (1 +- a_N)/2

where the +- are signs to be taken independently,

are 2^N primitive idempotents, a complete set that adds to 1 and mutually annihilate.

Now let V be a primitive idempotent that is taken from a mutually unbiased basis with respect to the above. For the definition of “mutually unbiased basis” see previous posts. Or one could require that

V A V = V / 2^N

for all A, whatever the choice of signs, and where N is the dimensionality of the Hilbert space.

Now let A and A’ be two different primitive idempotents from that complete set of 2^N generated by a_n. Then the product:

A V A’

is nilpotent (and non zero). The nilpotency follows because A A’ = 0. There are 2^N(2^N-1) of these things, each different.

One typically normalizes the nilpotent A V A’ by multiplying by 2^N. Then they satisfy a cool sum rule. The sum of all the 2^N(2^N-1) nilpotents, plus all the 2^N idempotents, is 2^N times V:

V = A + A’ + …

+ A V A’ + A’ V A” + …

Similarly, given a “complete set of mutually annihilating primitive nilpotents”, (which is a little harder to define than a complete set of mutually annihilating primitive idempotents), you can put together a complete set of idempotents if you have a primitive idempotent V that is unbiased with respect to the nilpotents. These definitions are different from the (to me, much more important) idempotent case, but if you look at examples from matrix nilpotents, it will be reasonably clear how to go about setting it up.

Physically, pure density matrices are idempotents, while creation and annihilation operators are nilpotents. The above conversions amount, in a certain way, to converting density matrices into creation and annihilation operators by having those operators act on the quantum vacuum. Algebraically, the quantum vacuum is the state V. The product AV is a creation operator, and the product VA is an annihilation operator.

These ideas are not entirely my own. See Julian Schwinger’s papers on the Measurement Algebra for the same thing, but not as nicely presented in that he was writing in the 1950s, and I think Clifford algebra makes it more natural. In his papers, M( a’ , a’ ) is the pure density matrix for the state a’, and he’s unaware of the mutually unbiased basis stuff (which really isn’t needed, but it makes for nice normalization, and it fits nicely with the other stuff going on around here).