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† 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 ΞzdagΞ=mI. The unitary zeon matrices of order n form a subgroup Un 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 ΞzdagΞ=mI+Γ, where Γ is a diagonal nilpotent zeon matrix. The weakly unitary matrices of order n form a group Wn, of which Un 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.
Related
- Asymptotic expansion of smooth functions in deterministic and iid Haar unitary matrices, and application to tensor products of matrices
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- On Trace Averages and Sharp Asymptotics for Powers of Haar-Unitary Matrices
BibTeX
@misc{staples2026weakly,
title = {{Weakly Unitary Zeon Matrices}},
author = {G. Stacey Staples},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33891},
eprint = {2610.33891},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33891},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Weakly Unitary Zeon Matrices
AU - G. Stacey Staples
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2610.33891
UR - https://arxiv.org/abs/2610.33891
AB - 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^† 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.
ER -