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A Spectral Theorem for Zeon Matrices

G. Stacey Staples

TL;DR

The paper develops a spectral theory for matrices over the complex zeon algebra $\mathbb{C}\mathfrak{Z}$, defining zeon eigenvalues as spectrally simple zeros of the zeon characteristic polynomial and connecting them to the eigenstructure of the scalar part $\mathfrak{C}A$. It proves a Zeon Spectral Theorem for self-adjoint, spectrally simple matrices, expressing $A$ as $A=\bigoplus_{j=1}^m \lambda_j\pi_j$ with rank-one projections $\pi_j$, and establishes a resolution of the identity via an orthonormal zeon basis. The framework yields practical finite-sum evaluations of matrix exponentials, including applications to nilpotent matrices and zeon Laplacians, and provides tools for diagonalization and graph- theoretic counting via zeon algebra. Together, these results extend classical spectral theory to the CZ-linear setting and deepen connections to combinatorics and graph theory through zeon algebraic techniques. The work lays a foundation for further study of zeon matrix norms, unitary/orthogonal zeon matrices, and additional CZ-analytic methods with potential wide-ranging applications.

Abstract

In this paper, spectral properties of matrices with (complex) zeon entries are investigated. It is shown that when $A$ is an $m\times m$ self-adjoint matrix whose characteristic polynomial $χ_A(u)$ has $m$ ``spectrally simple'' zeros $λ_1, \ldots, λ_m$ in the zeon algebra ${\mathbb{C}\mathfrak{Z}}$, there exist $m$ linearly independent normalized zeon eigenvectors $v_1, \ldots, v_m$ such that $A=\bigoplus_{j=1}^m λ_jπ_j$, where $π_j=v_j{v_j}^†$ is a rank-one projection onto the zeon submodule ${\rm span}\{v_j\}$ for $j=1, \ldots, m$.

A Spectral Theorem for Zeon Matrices

TL;DR

The paper develops a spectral theory for matrices over the complex zeon algebra , defining zeon eigenvalues as spectrally simple zeros of the zeon characteristic polynomial and connecting them to the eigenstructure of the scalar part . It proves a Zeon Spectral Theorem for self-adjoint, spectrally simple matrices, expressing as with rank-one projections , and establishes a resolution of the identity via an orthonormal zeon basis. The framework yields practical finite-sum evaluations of matrix exponentials, including applications to nilpotent matrices and zeon Laplacians, and provides tools for diagonalization and graph- theoretic counting via zeon algebra. Together, these results extend classical spectral theory to the CZ-linear setting and deepen connections to combinatorics and graph theory through zeon algebraic techniques. The work lays a foundation for further study of zeon matrix norms, unitary/orthogonal zeon matrices, and additional CZ-analytic methods with potential wide-ranging applications.

Abstract

In this paper, spectral properties of matrices with (complex) zeon entries are investigated. It is shown that when is an self-adjoint matrix whose characteristic polynomial has ``spectrally simple'' zeros in the zeon algebra , there exist linearly independent normalized zeon eigenvectors such that , where is a rank-one projection onto the zeon submodule for .
Paper Structure (21 sections, 19 theorems, 110 equations, 1 figure)

This paper contains 21 sections, 19 theorems, 110 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Zeon Annihilators
  4. $\mathbb{C}\mathfrak{Z}$-Linear Algebra
  5. Linear Independence in $\mathbb{C}\mathfrak{Z}^m$
  6. Zeon Matrices as $\mathbb{C}\mathfrak{Z}$-Linear Operators on $\mathbb{C}\mathfrak{Z}^m$
  7. The Kernel of a Zeon Matrix
  8. Eigenvalues, Eigenvectors, and the Characteristic Polynomial
  9. Spectrally Simple Zeros of Complex Zeon Polynomials
  10. Zeon Eigenvalues
  11. Zeon eigenvectors
  12. The Zeon Spectral Theorem
  13. The Inner Product on $\mathbb{C}\mathfrak{Z}^m$
  14. Zeon Matrix Exponentials as Finite Sums
  15. Self-Adjoint Matrices
  16. ...and 6 more sections

Key Result

Proposition 2.3

Let $u\in\mathbb{C}\mathfrak{Z}$, and let $\kappa$ denote the index of nilpotency In particular, $\kappa$ is the least positive integer such that $(\mathfrak{D}u)^\kappa=0$. of $\mathfrak{D}u$. It follows that $u$ is uniquely invertible if and only if $\mathfrak{C}\mspace{1mu} u\ne0$, and the invers

Figures (1)

  • Figure 1: An $8$-vertex graph and its nilpotent adjacency matrix.

Theorems & Definitions (55)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Definition 2.5
  • Example 2.6
  • Definition 2.7
  • Definition 2.8
  • Example 2.9
  • Lemma 2.10: Annihilator Ideals
  • ...and 45 more