Non-Expansive Matrix Based number Systems

TL;DR

This paper investigates minimal length representations of vectors using powers of a specific Jordan block matrix and extends previous work to larger Jordan blocks with eigenvalue -1.

Contribution

It answers a question about minimal representations for 2x2 Jordan blocks and generalizes the results to n x n Jordan blocks with eigenvalue -1.

Findings

Determined minimal length representations for 2x2 Jordan blocks.

Extended the analysis to n x n Jordan blocks with eigenvalue -1.

Provided a framework for understanding number systems based on matrix actions.

Abstract

Let $M = \left(\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}\right)$ be a $2 \times 2$ Jordan block with eigenvalue $1$, and let $\mathcal{D} = \{\left(\begin{smallmatrix}0 \\ 1 \end{smallmatrix}\right), \left(\begin{smallmatrix} 0 \\ -1 \end{smallmatrix} \right)\}$. In this paper, we answer a question of Caldwell, Hare, and V\'avra about the minimal length representation of $\left( \begin{smallmatrix} a \\ b \end{smallmatrix} \right) = \sum_{i=0}^{k-1} M^i d_i$ with $d_i \in \mathcal{D}$. Further, we extend the work of Caldwell, Hare, and V\'avra to consider the case of $n \times n$ Jordan blocks with eigenvalue $-1$.