A Spectral Theorem for Zeon Matrices

TL;DR

This paper establishes a spectral theorem for self-adjoint matrices with zeon entries, showing they can be diagonalized with eigenvectors and eigenvalues in the zeon algebra, extending classical spectral theory.

Contribution

It introduces a spectral theorem for zeon matrices, demonstrating diagonalization with eigenvectors and eigenvalues within the zeon algebra framework, a novel extension of classical results.

Findings

Existence of linearly independent zeon eigenvectors for self-adjoint zeon matrices.

Diagonalization of zeon matrices into a direct sum of eigenvalues and projections.

Spectral properties depend on zeros of the characteristic polynomial in the zeon algebra.

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 $\chi_A(u)$ has $m$ ``spectrally simple'' zeros $\lambda_1, \ldots, \lambda_m$ in the zeon algebra ${\mathbb{C}\mathfrak{Z}}$, there exist $m$ linearly independent normalized zeon eigenvectors $v_1, \ldots, v_m$ such that $A=\bigoplus_{j=1}^m \lambda_j\pi_j$, where $\pi_j=v_j{v_j}^\dag$ is a rank-one projection onto the zeon submodule ${\rm span}\{v_j\}$ for $j=1, \ldots, m$.