Weakly Unitary Zeon Matrices

Abstract

In this paper, zeon matrix groups are discussed for the first time. Extending the concept of unitary complex matrices (i.e., matrices $A$ such that $A^\dagger A=\mI$) to matrices having entries in the complex zeon algebra $\CZ$, a ``zeon dagger'' operator $\zdag$ is defined and used to declare a zeon matrix $Ξ$ to be {\em unitary} if $Ξ^\zdagΞ=\mI$. The unitary zeon matrices of order $n$ form a subgroup $\U_n$ of all invertible order-$n$ zeon matrices. A more interesting notion is that of ``weak unitarity'' of a zeon matrix. In particular, a zeon matrix $Ξ$ will be said to be {\em weakly unitary} if $Ξ^\zdagΞ=\mI+Γ$, where $Γ$ is a diagonal nilpotent zeon matrix. The weakly unitary matrices of order $n$ form a group $\W_n$, of which $\U_n$ is a subgroup. It is further shown that the exponential of a graph's nilpotent adjacency matrix has a natural orthogonal decomposition into the sum of a weakly unitary zeon matrix and a nilpotent zeon matrix. Eigenvalues of the weakly unitary zeon matrix reveal information about the graph's cycles and paths.