Non-Expansive Matrix Based number Systems

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