A Spectral Theorem for Zeon Matrices

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 ${\lambda }_1,\dots ,{\lambda }_m$ in the zeon algebra $\mathbb{C}\mathfrak{Z}$, there exist $m$ linearly independent normalized zeon eigenvectors $v_1,\dots ,v_m$ such that $A={⨁}_{j=1}^m{\lambda }_j{\pi }_j$, where ${\pi }_j=v_j{v_j}^{†}$ is a rank-one projection onto the zeon submodule $\mathrm{span}\{v_j\}$ for $j=1,\dots ,m$.