Kea recently wrote about Rowland’s being banned on arXiv. Rowland’s efforts are kind of similar to mine except that he works with the nilpotents (BB = 0) of a Clifford algebra while I work with the idempotents (BB = B). Some years ago, I had a conversation with Hestenes and discussed these things, and he mentioned that there is a close relationship between nilpotents and idempotents. Since then, I guess I’ve learned enough Clifford algebra to write about that relationship and so this post.
Let’s begin by picking up some intuition from the Pauli matrices, the well known matrix representation of the real Clifford algebra generated by three basis vectors with signature +++. As usual, we will call the three basis vectors x, y, z. The “presentation” of the algebra (i.e. the rules that distinguish it from the free algebra on x, y, and z) are that xx = yy = zz = 1, and that x, y, and z anticommute. The Pauli matrices are as follows:
A few idempotent and nilpotent 2×2 matrices are the following:
The above four are of interest because they exhaust the degrees of freedom of the 2×2 matrices. Of course there are an infinite number of idempotent 2×2 matrices and the same for the nilpotent. To get some intuition about the relationship between nilpotents and idempotents, write the above four matrices in terms of the Pauli spin matrices and unity. (The reader might be more familiar with this sort of thing in the Dirac algebra, where one often has cause for writing a 4×4 matrix in terms of the Dirac bilinears. This amounts to writing 4×4 matrices using a basis set of the 16 Dirac bilinears. The Pauli matrices are only 2×2 and so require a basis set of only four elements. We use the unit matrix and the three Pauli spin matrices.)