Fourier pairs

The anticommutativity condition can be extended in different directions. One is to look for more and more pairs which anticommute. That would lead to such things as quaternionsquaternionDirac matrixPauli matrix, Dirac matrices, and similar artifacts. Another is to take some other numerical factor, writing $M N = \omega N M$. Then there is a new alternative: $\omega$ would have to be a root of unity if $M$ and $N$ generated cycles of each others eigenvectors, or else one of the matrices would have to be singular to terminate the chain of eigenvectors, each with different eigenvalues. Or, finally, the space might not be finite dimensional.

Amongst all the possibilities, there is one which has an interesting symmetry. Suppose $\omega$ is the nth root of unity with smallest nonzero argument for n-dimensional matrices $M$ and $N$. Then following the same reasoning as before, these matrices could be brought to the associated forms $W$ and $S$,

$$\begin{eqnarray*} W & = & \left[ \begin{array}{ccccc} 1 & . & . & \ldots & ... ... & \ldots & . \\ 1 & . & . & \ldots & . \end{array} \right] \end{eqnarray*}$$

so any general matrix could be written in the form

$$\begin{eqnarray*} A & = & \sum{ a_{ij} W^i S^j }, \end{eqnarray*}$$

which is a Finite Fourier Series decomposition of $A$, in a manner of speaking.